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Но не все из них имеют физический смысл! Поэтому после решения уравнения всегда нужно проверять полученные ответы на соответствие реальности. 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Подставляем это значение в любое удобное уравнение выше, например в ",{"__TSPROSE_proseElement":174,"schema":1704,"data":1705,"storageKey":1707,"id":1708},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1706},"4a + 6 = p - 6","$ 4a + 6 = p - 6 $","inlinerMath-JdI8TDC4R",{"__TSPROSE_proseElement":174,"schema":1710,"data":1711},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},", и находим возраст Павла:","paragraph-Bp4yBVeKz",{"__TSPROSE_proseElement":174,"schema":1714,"data":1715,"storageKey":1717,"id":1718},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":1716,"freeze":241},"4 \\cdot 11 + 6 = p - 6 \\\\ 44 + 6 = p - 6 \\\\ 50 = p - 6 \\\\ p = 56","$$ 4 \\cdot 11 + 6 = p - 6 \\\\ 44 + 6 = p - 6 \\\\ 50 = p - 6 \\\\ p = 56 $$","blockMath-eIlP2obVQ",{"__TSPROSE_proseElement":174,"schema":1720,"children":1721,"id":1737},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[1722,1725,1729,1731,1735],{"__TSPROSE_proseElement":174,"schema":1723,"data":1724},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Итак, Павлу сейчас ",{"__TSPROSE_proseElement":174,"schema":1726,"data":1727,"storageKey":1469,"id":1728},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1433},"inlinerMath-WYRLesvjE-1",{"__TSPROSE_proseElement":174,"schema":1730,"data":1473},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":1732,"data":1733,"storageKey":1477,"id":1734},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1436},"inlinerMath-vFI6JrYG1-3",{"__TSPROSE_proseElement":174,"schema":1736,"data":531},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"paragraph-OCPlwK9jS","vozrastnye-problemy",{"__TSPROSE_proseElement":174,"schema":1740,"data":1741,"children":1745,"id":2137},{"name":332,"type":240,"linkable":174,"__TSPROSE_schema":174},{"info":1742},{"title":1743,"level":1412,"attributes":1744},"сАд перфекциониста",[],[1746,1790,1800,1821,1844],{"__TSPROSE_proseElement":174,"schema":1747,"children":1748},{"name":341,"type":240,"linkable":241,"__TSPROSE_schema":174},[1749],{"__TSPROSE_proseElement":174,"schema":1750,"children":1751,"id":1789},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[1752,1755,1759,1762,1768,1771,1777,1780,1786],{"__TSPROSE_proseElement":174,"schema":1753,"data":1754},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"В городском парке есть большая прямоугольная клумба размерами ",{"__TSPROSE_proseElement":174,"schema":1756,"data":1757,"storageKey":1077,"id":1758},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1076},"inlinerMath-8Y3Z60Vbg-2",{"__TSPROSE_proseElement":174,"schema":1760,"data":1761},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," метров на ",{"__TSPROSE_proseElement":174,"schema":1763,"data":1764,"storageKey":1766,"id":1767},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1765},"15","$ 15 $","inlinerMath-1xC295UiR",{"__TSPROSE_proseElement":174,"schema":1769,"data":1770},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," метров. 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Найдите, с каким расстоянием должны быть посажены цветы вдоль каждой из сторон клумбы.","paragraph-0KRdSgbPW",{"__TSPROSE_proseElement":174,"schema":1791,"data":1792},{"name":387,"type":240,"linkable":241,"__TSPROSE_schema":174},{"label":1793,"serializedValidator":1794},"Расстояние между цветами",{"type":1169,"ordered":241,"separator":1170,"answers":1795},[1796,1798],{"__ERUDIT_CHECK":174,"name":391,"data":1797},{"expr":393},{"__ERUDIT_CHECK":174,"name":391,"data":1799},{"expr":632},{"__TSPROSE_proseElement":174,"schema":1801,"children":1802},{"name":396,"type":240,"linkable":241,"__TSPROSE_schema":174},[1803],{"__TSPROSE_proseElement":174,"schema":1804,"children":1805,"id":1820},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[1806,1809,1817],{"__TSPROSE_proseElement":174,"schema":1807,"data":1808},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Для нахождения расстояния между цветами сторону клумбы надо делить на количество цветов, ",{"__TSPROSE_proseElement":174,"schema":1810,"data":1811,"children":1812,"id":1816},{"name":269,"type":250,"linkable":174,"__TSPROSE_schema":174},{"type":271},[1813],{"__TSPROSE_proseElement":174,"schema":1814,"data":1815},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"уменьшенное на единицу!","emphasis-xfaofL223",{"__TSPROSE_proseElement":174,"schema":1818,"data":1819},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," Потому что количество промежутков между цветами на стороне клумбы всегда на единицу меньше, чем количество цветов, посаженных вдоль этой стороны.","paragraph-yizxIYLRs",{"__TSPROSE_proseElement":174,"schema":1822,"children":1823},{"name":407,"type":240,"linkable":241,"__TSPROSE_schema":174},[1824],{"__TSPROSE_proseElement":174,"schema":1825,"children":1826,"id":1843},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[1827,1830,1834,1837,1841],{"__TSPROSE_proseElement":174,"schema":1828,"data":1829},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Расстояние между цветами вдоль большей стороны клумбы равно ",{"__TSPROSE_proseElement":174,"schema":1831,"data":1832,"storageKey":655,"id":1833},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":632},"inlinerMath-mqn34wvB0-4",{"__TSPROSE_proseElement":174,"schema":1835,"data":1836},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," метра, а вдоль меньшей стороны -- ",{"__TSPROSE_proseElement":174,"schema":1838,"data":1839,"storageKey":418,"id":1840},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":393},"inlinerMath-z9Yar9waf-6",{"__TSPROSE_proseElement":174,"schema":1842,"data":935},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"paragraph-HtS2HPAWX",{"__TSPROSE_proseElement":174,"schema":1845,"children":1846},{"name":426,"type":240,"linkable":241,"__TSPROSE_schema":174},[1847,1878,1885,1891,1898,1919,1925,1962,1968,1984,1990,1996,2002,2014,2020,2042,2048,2085,2091,2105,2111],{"__TSPROSE_proseElement":174,"schema":1848,"children":1849,"id":1877},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[1850,1853,1857,1860,1866,1869,1875],{"__TSPROSE_proseElement":174,"schema":1851,"data":1852},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Обозначим за ",{"__TSPROSE_proseElement":174,"schema":1854,"data":1855,"storageKey":438,"id":1856},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":437},"inlinerMath-TLSwLL8iH-6",{"__TSPROSE_proseElement":174,"schema":1858,"data":1859},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," количество цветов вдоль большей стороны клумбы, а за ",{"__TSPROSE_proseElement":174,"schema":1861,"data":1862,"storageKey":1864,"id":1865},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1863},"y","$ y $","inlinerMath-GydTYC7Nc",{"__TSPROSE_proseElement":174,"schema":1867,"data":1868},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," количество цветов вдоль меньшей стороны клумбы. Произведение этих чисел должно быть равно количеству цветов, то есть ",{"__TSPROSE_proseElement":174,"schema":1870,"data":1871,"storageKey":1873,"id":1874},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1872},"xy = 36","$ xy = 36 $","inlinerMath-eyJl4CLbq",{"__TSPROSE_proseElement":174,"schema":1876,"data":531},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"paragraph-VVy83v6iV",{"__TSPROSE_proseElement":174,"schema":1879,"children":1880,"id":1884},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[1881],{"__TSPROSE_proseElement":174,"schema":1882,"data":1883},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Рассчитаем расстояние между цветами вдоль каждой стороны клумбы:","paragraph-eMZv281L2",{"__TSPROSE_proseElement":174,"schema":1886,"data":1887,"storageKey":1889,"id":1890},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":1888,"freeze":241},"\\frac{15}{x-1} >>{big} \\frac{10}{y-1}","$$ \\frac{15}{x-1} >>{big} \\frac{10}{y-1} $$","blockMath-1NIIXTAOx",{"__TSPROSE_proseElement":174,"schema":1892,"children":1893,"id":1897},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[1894],{"__TSPROSE_proseElement":174,"schema":1895,"data":1896},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Единицу мы вычитаем, потому что количество промежутков между цветами на стороне клумбы всегда на единицу меньше, чем количество цветов, посаженных вдоль этой стороны. Например, между двумя цветками всего один промежуток!","paragraph-MutP2wDnm",{"__TSPROSE_proseElement":174,"schema":1899,"children":1900,"id":1918},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[1901,1904,1908,1911,1915],{"__TSPROSE_proseElement":174,"schema":1902,"data":1903},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"По условию нам известно, что большее расстояние (которое с ",{"__TSPROSE_proseElement":174,"schema":1905,"data":1906,"storageKey":438,"id":1907},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":437},"inlinerMath-TLSwLL8iH-7",{"__TSPROSE_proseElement":174,"schema":1909,"data":1910},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},") на ",{"__TSPROSE_proseElement":174,"schema":1912,"data":1913,"storageKey":1784,"id":1914},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1783},"inlinerMath-GWJ3eIxvK-1",{"__TSPROSE_proseElement":174,"schema":1916,"data":1917},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," метр больше, чем меньшее расстояние. Это можно записать в виде равенства:","paragraph-6EZDWm9xm",{"__TSPROSE_proseElement":174,"schema":1920,"data":1921,"storageKey":1923,"id":1924},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":1922,"freeze":241},"\\frac{15}{x-1} = \\frac{10}{y-1} + 1","$$ \\frac{15}{x-1} = \\frac{10}{y-1} + 1 $$","blockMath-mWTPIiVfu",{"__TSPROSE_proseElement":174,"schema":1926,"children":1927,"id":1961},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[1928,1931,1935,1938,1942,1945,1949,1952,1958],{"__TSPROSE_proseElement":174,"schema":1929,"data":1930},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Из соотношения ",{"__TSPROSE_proseElement":174,"schema":1932,"data":1933,"storageKey":1873,"id":1934},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1872},"inlinerMath-eyJl4CLbq-1",{"__TSPROSE_proseElement":174,"schema":1936,"data":1937},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," выражаем ",{"__TSPROSE_proseElement":174,"schema":1939,"data":1940,"storageKey":1864,"id":1941},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1863},"inlinerMath-GydTYC7Nc-1",{"__TSPROSE_proseElement":174,"schema":1943,"data":1944},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," через ",{"__TSPROSE_proseElement":174,"schema":1946,"data":1947,"storageKey":438,"id":1948},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":437},"inlinerMath-TLSwLL8iH-8",{"__TSPROSE_proseElement":174,"schema":1950,"data":1951},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," -- ",{"__TSPROSE_proseElement":174,"schema":1953,"data":1954,"storageKey":1956,"id":1957},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1955},"y = \\frac{36}{x}","$ y = \\frac{36}{x} $","inlinerMath-k4fFCUP9S",{"__TSPROSE_proseElement":174,"schema":1959,"data":1960},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},". 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Но расстояния между ними будут разными. 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Причем достаточно примеров как реальных проблем, так и всевозможных \"загадок\" с хитро поставленными условиями.","paragraph-qeFPceXxI",{"__TSPROSE_proseElement":174,"schema":2151,"data":2152,"id":2155},{"name":208,"type":240,"linkable":174,"__TSPROSE_schema":174},{"level":2153,"title":2154},2,"Движение с ускорением","dvizhenie-s-uskoreniem",{"__TSPROSE_proseElement":174,"schema":2157,"children":2158,"id":2179},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[2159,2162,2170,2173,2177],{"__TSPROSE_proseElement":174,"schema":2160,"data":2161},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Когда на тела действуют какие-то силы, их скорость меняется: пинок футбольного мяча ногой, бросок снежка рукой, тетива лука, выталкивающая стрелу или даже притяжение Земли, разгоняющее падающий вниз молоток. Это изменение скорости можно померить. \"Скорость, с которой меняется скорость\", как бы непривычно это не звучало, называется ",{"__TSPROSE_proseElement":174,"schema":2163,"data":2164,"children":2165,"id":2169},{"name":269,"type":250,"linkable":174,"__TSPROSE_schema":174},{"type":271},[2166],{"__TSPROSE_proseElement":174,"schema":2167,"data":2168},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"ускорением","emphasis-ZJSY91USZ",{"__TSPROSE_proseElement":174,"schema":2171,"data":2172},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," и обозначается буквой ",{"__TSPROSE_proseElement":174,"schema":2174,"data":2175,"storageKey":1504,"id":2176},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1503},"inlinerMath-YqDfaaRXA-1",{"__TSPROSE_proseElement":174,"schema":2178,"data":531},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"paragraph-64BBegjZE",{"__TSPROSE_proseElement":174,"schema":2181,"children":2182,"id":2231},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[2183,2186,2192,2195,2201,2204,2210,2213,2219,2222,2228],{"__TSPROSE_proseElement":174,"schema":2184,"data":2185},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Отличная новость состоит в том, что простые виды движения с ускорением напрямую описываются квадратными уравнениями! Даже никаких хитрых условий не нужно, чтобы они появились! Классический пример -- свободное падение под действием силы притяжения Земли. Наша планета любой объект разгоняет дополнительно на ",{"__TSPROSE_proseElement":174,"schema":2187,"data":2188,"storageKey":2190,"id":2191},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":2189},"g = 10","$ g = 10 $","inlinerMath-EMd7i6dpk",{"__TSPROSE_proseElement":174,"schema":2193,"data":2194},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," метров в секунду за каждую секунду, пока тот находится в воздухе! 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Скорее всего, это был ",{"__TSPROSE_proseElement":174,"schema":2637,"data":2638,"children":2639,"id":2643},{"name":269,"type":250,"linkable":174,"__TSPROSE_schema":174},{"type":271},[2640],{"__TSPROSE_proseElement":174,"schema":2641,"data":2642},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"мысленный эксперимент","emphasis-PJcYJrIgH",{"__TSPROSE_proseElement":174,"schema":2645,"data":2646},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},": если тяжелый предмет падает быстрее легкого, то что произойдет, если связать их веревкой? С одной стороны, легкий предмет должен тормозить тяжелый. С другой стороны, вместе они весят ещё больше, чем тяжелый сам по себе, то есть должны упасть ещё быстрее -- противоречие! Из этого Галилей заключил, что все тела падают за одно и то же время.","paragraph-ZPyGxTq25",{"__TSPROSE_proseElement":174,"schema":2649,"children":2650,"id":2663},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[2651,2654,2660],{"__TSPROSE_proseElement":174,"schema":2652,"data":2653},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"А вот настоящий эксперимент провёл астронавт Дэвид Скотт, причем не на Пизанской башне, а сразу на Луне! Во время миссии \"Аполлон-15\" в ",{"__TSPROSE_proseElement":174,"schema":2655,"data":2656,"storageKey":2658,"id":2659},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":2657},"1971","$ 1971 $","inlinerMath-XXY2DX4Wp",{"__TSPROSE_proseElement":174,"schema":2661,"data":2662},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," году он одновременно выпустил из рук молоток и перо. Они упали на лунную поверхность одновременно, наглядно продемонстрировав, что без сопротивления воздуха все тела падают за одно и то же время, независимо от их массы.","paragraph-6W3bgslj0",{"__TSPROSE_proseElement":174,"schema":2665,"data":2667,"storageKey":2668,"id":2670},{"name":2666,"type":240,"linkable":174,"__TSPROSE_schema":174},"video",{"src":2668,"autoplay":174,"width":2669},"content\u002F01-foundations\u002F02-equations\u002F03-quadratic\u002F6-real-world\u002Fassets\u002Fhammer-feather.mp4","350px","video-CS3hSMphw","broski-sharov-s-pizanskoy-bashni",{"__TSPROSE_proseElement":174,"schema":2673,"data":2674,"id":2676},{"name":208,"type":240,"linkable":174,"__TSPROSE_schema":174},{"level":2153,"title":2675},"Электричество","elektrichestvo",{"__TSPROSE_proseElement":174,"schema":2678,"children":2679,"id":2710},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[2680,2683,2689,2692,2698,2701,2707],{"__TSPROSE_proseElement":174,"schema":2681,"data":2682},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Квадратные уравнения пробрались и в электрические цепи. Элементы цепи можно соединять последовательно и параллельно, что позволяет гибко управлять напряжением, силой тока и сопротивлением. При параллельном соединении общее сопротивление в цепи ",{"__TSPROSE_proseElement":174,"schema":2684,"data":2685,"storageKey":2687,"id":2688},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":2686},"R_t","$ R_t $","inlinerMath-I2v9y0zZy",{"__TSPROSE_proseElement":174,"schema":2690,"data":2691},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," рассчитывается по хитрой формуле через сопротивления отдельных элементов ",{"__TSPROSE_proseElement":174,"schema":2693,"data":2694,"storageKey":2696,"id":2697},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":2695},"R_1","$ R_1 $","inlinerMath-BvgC7BoEN",{"__TSPROSE_proseElement":174,"schema":2699,"data":2700},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},", ",{"__TSPROSE_proseElement":174,"schema":2702,"data":2703,"storageKey":2705,"id":2706},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":2704},"R_2","$ R_2 $","inlinerMath-uYvjK2nti",{"__TSPROSE_proseElement":174,"schema":2708,"data":2709},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," и так далее:","paragraph-FD0qHZyaN",{"__TSPROSE_proseElement":174,"schema":2712,"data":2713,"storageKey":2715,"id":2716},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":2714,"freeze":241},"\\frac{1}{R_t} = \\frac{1}{R_1} + \\frac{1}{R_2} + \\dots","$$ \\frac{1}{R_t} = \\frac{1}{R_1} + \\frac{1}{R_2} + \\dots $$","blockMath-lxOOF58Mn",{"__TSPROSE_proseElement":174,"schema":2718,"data":2719,"children":2723,"id":2967},{"name":332,"type":240,"linkable":174,"__TSPROSE_schema":174},{"info":2720},{"title":2721,"level":336,"attributes":2722},"Ремонт обогревателя",[],[2724,2754,2764,2774,2798],{"__TSPROSE_proseElement":174,"schema":2725,"children":2726},{"name":341,"type":240,"linkable":241,"__TSPROSE_schema":174},[2727,2748],{"__TSPROSE_proseElement":174,"schema":2728,"children":2729,"id":2747},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[2730,2733,2737,2740,2744],{"__TSPROSE_proseElement":174,"schema":2731,"data":2732},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Мастеру принесли на ремонт электрический обогреватель, который состоит из двух параллельно соединенных нагревательных элементов. Все элементы обогревателя залиты эпоксидной смолой, поэтому измерить сопротивление каждого элемента по отдельности невозможно, но клиент сказал, что сопротивление одного элемента на ",{"__TSPROSE_proseElement":174,"schema":2734,"data":2735,"storageKey":655,"id":2736},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":632},"inlinerMath-mqn34wvB0-10",{"__TSPROSE_proseElement":174,"schema":2738,"data":2739},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," Ом больше, чем сопротивление другого. Мастер измерил общее сопротивление цепи -- ",{"__TSPROSE_proseElement":174,"schema":2741,"data":2742,"storageKey":418,"id":2743},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":393},"inlinerMath-z9Yar9waf-8",{"__TSPROSE_proseElement":174,"schema":2745,"data":2746},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," Ом. Найдите сопротивление каждого элемента обогревателя.","paragraph-FtIEFfEY9",{"__TSPROSE_proseElement":174,"schema":2749,"data":2750,"storageKey":2751,"id":2753},{"name":380,"type":240,"linkable":174,"__TSPROSE_schema":174},{"src":2751,"width":624,"invert":2752},"content\u002F01-foundations\u002F02-equations\u002F03-quadratic\u002F6-real-world\u002Fassets\u002Fcircuit.svg","dark","image-QSX3w0pRK",{"__TSPROSE_proseElement":174,"schema":2755,"data":2756},{"name":387,"type":240,"linkable":241,"__TSPROSE_schema":174},{"label":2757,"serializedValidator":2758},"Сопротивления элементов",{"type":1169,"ordered":241,"separator":1170,"answers":2759},[2760,2762],{"__ERUDIT_CHECK":174,"name":391,"data":2761},{"expr":632},{"__ERUDIT_CHECK":174,"name":391,"data":2763},{"expr":613},{"__TSPROSE_proseElement":174,"schema":2765,"children":2766},{"name":396,"type":240,"linkable":241,"__TSPROSE_schema":174},[2767],{"__TSPROSE_proseElement":174,"schema":2768,"children":2769,"id":2773},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[2770],{"__TSPROSE_proseElement":174,"schema":2771,"data":2772},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"В виде равенства запишите сопротивление элемента с большим сопротивлением через сопротивление элемента с меньшим. Подставьте это выражение в формулу для параллельного соединения вместе с известным общим сопротивлением.","paragraph-PBMm3fEY1",{"__TSPROSE_proseElement":174,"schema":2775,"children":2776},{"name":407,"type":240,"linkable":241,"__TSPROSE_schema":174},[2777],{"__TSPROSE_proseElement":174,"schema":2778,"children":2779,"id":2797},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[2780,2783,2787,2790,2794],{"__TSPROSE_proseElement":174,"schema":2781,"data":2782},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Сопротивление первого элемента обогревателя равно ",{"__TSPROSE_proseElement":174,"schema":2784,"data":2785,"storageKey":655,"id":2786},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":632},"inlinerMath-mqn34wvB0-11",{"__TSPROSE_proseElement":174,"schema":2788,"data":2789},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," Ом, а сопротивление второго элемента равно ",{"__TSPROSE_proseElement":174,"schema":2791,"data":2792,"storageKey":614,"id":2793},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":613},"inlinerMath-9rSWhNw25-3",{"__TSPROSE_proseElement":174,"schema":2795,"data":2796},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," Ом.","paragraph-lHo4BjyCb",{"__TSPROSE_proseElement":174,"schema":2799,"children":2800},{"name":426,"type":240,"linkable":241,"__TSPROSE_schema":174},[2801,2815,2821,2835,2841,2860,2866,2905,2911,2943,2949],{"__TSPROSE_proseElement":174,"schema":2802,"children":2803,"id":2814},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[2804,2807,2811],{"__TSPROSE_proseElement":174,"schema":2805,"data":2806},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Клиент сказал, что сопротивление одного элемента (назовем его \"вторым\") на ",{"__TSPROSE_proseElement":174,"schema":2808,"data":2809,"storageKey":655,"id":2810},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":632},"inlinerMath-mqn34wvB0-12",{"__TSPROSE_proseElement":174,"schema":2812,"data":2813},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," Ом больше, чем сопротивление другого (\"первого\"). Запишем это в виде равенства:","paragraph-TlasnhAoF",{"__TSPROSE_proseElement":174,"schema":2816,"data":2817,"storageKey":2819,"id":2820},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":2818,"freeze":241},"R_2 = R_1 + 3","$$ R_2 = R_1 + 3 $$","blockMath-xP4CTHE6D",{"__TSPROSE_proseElement":174,"schema":2822,"children":2823,"id":2834},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[2824,2827,2831],{"__TSPROSE_proseElement":174,"schema":2825,"data":2826},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Мастер измерил общее сопротивление цепи -- ",{"__TSPROSE_proseElement":174,"schema":2828,"data":2829,"storageKey":418,"id":2830},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":393},"inlinerMath-z9Yar9waf-9",{"__TSPROSE_proseElement":174,"schema":2832,"data":2833},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," Ом. Используем формулу для параллельного соединения, чтобы записать уравнение для общего сопротивления сети:","paragraph-WGD3sgNfc",{"__TSPROSE_proseElement":174,"schema":2836,"data":2837,"storageKey":2839,"id":2840},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":2838,"freeze":241},"\\frac{1}{2} = \\frac{1}{R_1} + \\frac{1}{R_2}","$$ \\frac{1}{2} = \\frac{1}{R_1} + \\frac{1}{R_2} $$","blockMath-tr8SNBaUb",{"__TSPROSE_proseElement":174,"schema":2842,"children":2843,"id":2859},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[2844,2847,2851,2853,2857],{"__TSPROSE_proseElement":174,"schema":2845,"data":2846},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Подставляем в это уравнение выражение для ",{"__TSPROSE_proseElement":174,"schema":2848,"data":2849,"storageKey":2705,"id":2850},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":2704},"inlinerMath-uYvjK2nti-1",{"__TSPROSE_proseElement":174,"schema":2852,"data":1944},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":2854,"data":2855,"storageKey":2696,"id":2856},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":2695},"inlinerMath-BvgC7BoEN-1",{"__TSPROSE_proseElement":174,"schema":2858,"data":449},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"paragraph-MKPZeOn1a",{"__TSPROSE_proseElement":174,"schema":2861,"data":2862,"storageKey":2864,"id":2865},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":2863,"freeze":241},"\\frac{1}{2} = \\frac{1}{R_1} + \\frac{1}{R_1 + 3} \\\\ R_1(R_1 + 3) = 2(R_1 + 3) + 2R_1 \\\\ R_1^2 + 3R_1 = 2R_1 + 6 + 2R_1 \\\\ R_1^2 + 3R_1 = 4R_1 + 6 \\\\ R_1^2 - R_1 - 6 = 0","$$ \\frac{1}{2} = \\frac{1}{R_1} + \\frac{1}{R_1 + 3} \\\\ R_1(R_1 + 3) = 2(R_1 + 3) + 2R_1 \\\\ R_1^2 + 3R_1 = 2R_1 + 6 + 2R_1 \\\\ R_1^2 + 3R_1 = 4R_1 + 6 \\\\ R_1^2 - R_1 - 6 = 0 $$","blockMath-MIlDuivHG",{"__TSPROSE_proseElement":174,"schema":2867,"children":2868,"id":2904},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[2869,2872,2876,2878,2882,2884,2890,2892,2896,2898,2902],{"__TSPROSE_proseElement":174,"schema":2870,"data":2871},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Получили квадратное уравнение. Его можно ",{"__TSPROSE_proseElement":174,"schema":2873,"data":2874,"storageKey":491,"id":2875},{"name":488,"type":250,"linkable":174,"__TSPROSE_schema":174},{"label":490},"dependencyInliner-bpWwR9Iwp-3",{"__TSPROSE_proseElement":174,"schema":2877,"data":785},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":2879,"data":2880,"storageKey":2546,"id":2881},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":2545},"inlinerMath-DLMG5r2qd-1",{"__TSPROSE_proseElement":174,"schema":2883,"data":504},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":2885,"data":2886,"storageKey":2888,"id":2889},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":2887},"-6","$ -6 $","inlinerMath-6kVj401tO",{"__TSPROSE_proseElement":174,"schema":2891,"data":513},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":2893,"data":2894,"storageKey":418,"id":2895},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":393},"inlinerMath-z9Yar9waf-10",{"__TSPROSE_proseElement":174,"schema":2897,"data":522},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":2899,"data":2900,"storageKey":814,"id":2901},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":813},"inlinerMath-nfmGVV06J-2",{"__TSPROSE_proseElement":174,"schema":2903,"data":531},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"paragraph-DlQyrFK09",{"__TSPROSE_proseElement":174,"schema":2906,"data":2907,"storageKey":2909,"id":2910},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":2908,"freeze":241},"(R_1 - 3)(R_1 + 2) = 0","$$ (R_1 - 3)(R_1 + 2) = 0 $$","blockMath-NLkCHp4An",{"__TSPROSE_proseElement":174,"schema":2912,"children":2913,"id":2942},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[2914,2917,2923,2926,2932,2935,2939],{"__TSPROSE_proseElement":174,"schema":2915,"data":2916},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Получаем два варианта сопротивления первого элемента: ",{"__TSPROSE_proseElement":174,"schema":2918,"data":2919,"storageKey":2921,"id":2922},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":2920},"R_1 = 3","$ R_1 = 3 $","inlinerMath-Zf0cfVIqT",{"__TSPROSE_proseElement":174,"schema":2924,"data":2925},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," Ом или ",{"__TSPROSE_proseElement":174,"schema":2927,"data":2928,"storageKey":2930,"id":2931},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":2929},"R_1 = -2","$ R_1 = -2 $","inlinerMath-pF6Kn5kuv",{"__TSPROSE_proseElement":174,"schema":2933,"data":2934},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," Ом. Отрицательное сопротивление не имеет физического смысла, поэтому ",{"__TSPROSE_proseElement":174,"schema":2936,"data":2937,"storageKey":2921,"id":2938},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":2920},"inlinerMath-Zf0cfVIqT-1",{"__TSPROSE_proseElement":174,"schema":2940,"data":2941},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," Ом. 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Всем, даже хейтерам физики, знакома простая формула ",{"__TSPROSE_proseElement":174,"schema":2980,"data":2981,"storageKey":2983,"id":2984},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":2982},"S = V \\cdot t","$ S = V \\cdot t $","inlinerMath-v58EzLeWz",{"__TSPROSE_proseElement":174,"schema":2986,"data":2987},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},", которая связывает расстояние, скорость и время. 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Отрицательная скорость ",{"__TSPROSE_proseElement":174,"schema":3307,"data":3309,"storageKey":3311,"id":3312},{"name":3308,"type":250,"linkable":174,"__TSPROSE_schema":174},"referenceInliner",{"label":3310},"не имеет смысла","\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Freal-world\u002F$_whyNotReal","referenceInliner-71IyxN4GN",{"__TSPROSE_proseElement":174,"schema":3314,"data":3315},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," в данной задаче, поэтому скорость парома равна ",{"__TSPROSE_proseElement":174,"schema":3317,"data":3318,"storageKey":614,"id":3319},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":613},"inlinerMath-9rSWhNw25-7",{"__TSPROSE_proseElement":174,"schema":3321,"data":3322},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," км\u002Fч.","paragraph-gmlLKUlbw",{"__TSPROSE_proseElement":174,"schema":3325,"data":3328,"children":3330,"id":3345},{"name":3326,"type":240,"linkable":3327,"__TSPROSE_schema":174},"details","always",{"title":3329},"Отрицательная скорость не имеет смысла",[3331,3338],{"__TSPROSE_proseElement":174,"schema":3332,"children":3333,"id":3337},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[3334],{"__TSPROSE_proseElement":174,"schema":3335,"data":3336},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Когда мы переносим физическую задачу про окружающий мир на язык математики, то мы теряем ограничения, которые накладывает этот самый реальный мир. Поэтому в результате мы можем вместе с \"правильными\" ответами получить и дополнительные варианты, которые не имеют смысла в реальной жизни. Например отрицательные скорости, отрицательное время или отрицательные расстояния.","paragraph-Aq97mkBXc",{"__TSPROSE_proseElement":174,"schema":3339,"children":3340,"id":3344},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[3341],{"__TSPROSE_proseElement":174,"schema":3342,"data":3343},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Такие варианты нужно отбрасывать, так как они невоспроизводимы в реальном мире и могут \"существовать\" только в математических абстракциях.","paragraph-H9eW1peWz","why-not-real",{"__TSPROSE_proseElement":174,"schema":3347,"children":3348,"id":3352},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[3349],{"__TSPROSE_proseElement":174,"schema":3350,"data":3351},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Теперь найдём время, которое паром плыл против течения. Подставим найденную скорость в формулу времени против течения:","paragraph-8onPEwrX4",{"__TSPROSE_proseElement":174,"schema":3354,"data":3355,"storageKey":3357,"id":3358},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":3356,"freeze":241},"t_{\\rarr} = \\frac{8}{6 - 2} = \\frac{8}{4} = 2","$$ t_{\\rarr} = \\frac{8}{6 - 2} = \\frac{8}{4} = 2 $$","blockMath-nLwWdNI6N",{"__TSPROSE_proseElement":174,"schema":3360,"children":3361,"id":3371},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[3362,3365,3369],{"__TSPROSE_proseElement":174,"schema":3363,"data":3364},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Значит, паром плыл против течения ",{"__TSPROSE_proseElement":174,"schema":3366,"data":3367,"storageKey":418,"id":3368},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":393},"inlinerMath-z9Yar9waf-13",{"__TSPROSE_proseElement":174,"schema":3370,"data":3117},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"paragraph-EO7JKWdxw","ekskursiya-na-parome",{"__TSPROSE_proseElement":174,"schema":3374,"data":3375,"children":3378,"id":4142},{"name":190,"type":240,"linkable":174,"__TSPROSE_schema":174},{"title":3376,"level":1133,"attributes":3377},"В поисках скорости",[],[3379,3675,3977],{"__TSPROSE_proseElement":174,"schema":3380,"data":3382,"children":3383},{"name":3381,"type":240,"linkable":241,"__TSPROSE_schema":174},"subProblem",{},[3384,3423,3429,3458,3474],{"__TSPROSE_proseElement":174,"schema":3385,"children":3386},{"name":341,"type":240,"linkable":241,"__TSPROSE_schema":174},[3387],{"__TSPROSE_proseElement":174,"schema":3388,"children":3389,"id":3422},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[3390,3393,3397,3400,3406,3409,3413,3416,3420],{"__TSPROSE_proseElement":174,"schema":3391,"data":3392},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Турист проплыл на байдарке ",{"__TSPROSE_proseElement":174,"schema":3394,"data":3395,"storageKey":1766,"id":3396},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1765},"inlinerMath-1xC295UiR-1",{"__TSPROSE_proseElement":174,"schema":3398,"data":3399},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," километров против течения и ",{"__TSPROSE_proseElement":174,"schema":3401,"data":3402,"storageKey":3404,"id":3405},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":3403},"14","$ 14 $","inlinerMath-EG1ZwIV76",{"__TSPROSE_proseElement":174,"schema":3407,"data":3408},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," километров по течению, затратив на это столько же времени, сколько он потратил бы на то, чтобы проплыть ",{"__TSPROSE_proseElement":174,"schema":3410,"data":3411,"storageKey":363,"id":3412},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":362},"inlinerMath-W23zkylLw-1",{"__TSPROSE_proseElement":174,"schema":3414,"data":3415},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," километров в стоячей воде. Найдите собственную скорость байдарки, если скорость течения ",{"__TSPROSE_proseElement":174,"schema":3417,"data":3418,"storageKey":1784,"id":3419},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1783},"inlinerMath-GWJ3eIxvK-3",{"__TSPROSE_proseElement":174,"schema":3421,"data":3322},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"paragraph-WSB7mGbrG",{"__TSPROSE_proseElement":174,"schema":3424,"data":3425},{"name":387,"type":240,"linkable":241,"__TSPROSE_schema":174},{"label":3426,"serializedValidator":3427},"Собственная скорость байдарки",{"__ERUDIT_CHECK":174,"name":391,"data":3428},{"expr":613},{"__TSPROSE_proseElement":174,"schema":3430,"children":3431},{"name":396,"type":240,"linkable":241,"__TSPROSE_schema":174},[3432],{"__TSPROSE_proseElement":174,"schema":3433,"children":3434,"id":3457},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[3435,3438,3446,3449,3455],{"__TSPROSE_proseElement":174,"schema":3436,"data":3437},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Ключевой в этой задаче является фраза ",{"__TSPROSE_proseElement":174,"schema":3439,"data":3440,"children":3441,"id":3445},{"name":269,"type":250,"linkable":174,"__TSPROSE_schema":174},{"type":271},[3442],{"__TSPROSE_proseElement":174,"schema":3443,"data":3444},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"\"столько же времени\"","emphasis-s0tPIavns",{"__TSPROSE_proseElement":174,"schema":3447,"data":3448},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},". Она позволяет записать равенство, которое связывает две ситуации: \"время поездки туда-обратно с имеющимся течением\" = \"время поездки в стоячей воде\". Время можно выразить через расстояние и скорость, используя формулу ",{"__TSPROSE_proseElement":174,"schema":3450,"data":3451,"storageKey":3453,"id":3454},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":3452},"S = Vt","$ S = Vt $","inlinerMath-hns4rdECL",{"__TSPROSE_proseElement":174,"schema":3456,"data":531},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"paragraph-xV2gntw0r",{"__TSPROSE_proseElement":174,"schema":3459,"children":3460},{"name":407,"type":240,"linkable":241,"__TSPROSE_schema":174},[3461],{"__TSPROSE_proseElement":174,"schema":3462,"children":3463,"id":3473},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[3464,3467,3471],{"__TSPROSE_proseElement":174,"schema":3465,"data":3466},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Собственная скорость байдарки равна ",{"__TSPROSE_proseElement":174,"schema":3468,"data":3469,"storageKey":614,"id":3470},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":613},"inlinerMath-9rSWhNw25-8",{"__TSPROSE_proseElement":174,"schema":3472,"data":3322},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"paragraph-zfqeTBIAH",{"__TSPROSE_proseElement":174,"schema":3475,"children":3476},{"name":426,"type":240,"linkable":241,"__TSPROSE_schema":174},[3477,3494,3500,3507,3512,3526,3532,3539,3545,3581,3587,3601,3607,3620,3626,3640,3646],{"__TSPROSE_proseElement":174,"schema":3478,"children":3479,"id":3493},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[3480,3483,3490],{"__TSPROSE_proseElement":174,"schema":3481,"data":3482},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Важно зацепиться за фразу ",{"__TSPROSE_proseElement":174,"schema":3484,"data":3485,"children":3486,"id":3489},{"name":269,"type":250,"linkable":174,"__TSPROSE_schema":174},{"type":271},[3487],{"__TSPROSE_proseElement":174,"schema":3488,"data":3444},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"emphasis-s0tPIavns-1",{"__TSPROSE_proseElement":174,"schema":3491,"data":3492},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},", то есть само время мы не знаем, но знаем, что для обеих ситуаций оно одинаковое. Этот факт позволяет нам приравнять друг к другу \"время поездки туда-обратно с имеющимся течением\" (ситуация первая) и \"время поездки в стоячей воде\" (ситуация вторая):","paragraph-rPSYBJwmf",{"__TSPROSE_proseElement":174,"schema":3495,"data":3496,"storageKey":3498,"id":3499},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":3497,"freeze":241},"\\boxed{t_{\\text{против течения}} + t_{\\text{по течению}} = t_{\\text{стоячая}}}","$$ \\boxed{t_{\\text{против течения}} + t_{\\text{по течению}} = t_{\\text{стоячая}}} $$","blockMath-8kHFEql3Y",{"__TSPROSE_proseElement":174,"schema":3501,"children":3502,"id":3506},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[3503],{"__TSPROSE_proseElement":174,"schema":3504,"data":3505},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Однако ничего про время нам из условия задачи неизвестно. Зато известны расстояния и скорости. Нам нужно выразить время через эти величины. Записываем классическую формулу, которая связывает время, скорость и расстояние:","paragraph-htCAdrkI9",{"__TSPROSE_proseElement":174,"schema":3508,"data":3509,"storageKey":3510,"id":3511},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":3452,"freeze":241},"$$ S = Vt $$","blockMath-d9Kkt7CCh",{"__TSPROSE_proseElement":174,"schema":3513,"children":3514,"id":3525},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[3515,3518,3522],{"__TSPROSE_proseElement":174,"schema":3516,"data":3517},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Разделим обе части равенства на ",{"__TSPROSE_proseElement":174,"schema":3519,"data":3520,"storageKey":3132,"id":3521},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":3131},"inlinerMath-zPgGLZYmJ-1",{"__TSPROSE_proseElement":174,"schema":3523,"data":3524},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},", чтобы оставить время в одиночестве с одной стороны равенства:","paragraph-MAHDUyxJ8",{"__TSPROSE_proseElement":174,"schema":3527,"data":3528,"storageKey":3530,"id":3531},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":3529,"freeze":241},"\\frac{S}{V} = \\frac{\\cancel{V}t}{\\cancel{V}} >>{big} \\boxed{t = \\frac{S}{V}}","$$ \\frac{S}{V} = \\frac{\\cancel{V}t}{\\cancel{V}} >>{big} \\boxed{t = \\frac{S}{V}} $$","blockMath-TrEit9oi5",{"__TSPROSE_proseElement":174,"schema":3533,"children":3534,"id":3538},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[3535],{"__TSPROSE_proseElement":174,"schema":3536,"data":3537},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Заменяем время в нашем равенстве для задачи:","paragraph-4sV3SsZxS",{"__TSPROSE_proseElement":174,"schema":3540,"data":3541,"storageKey":3543,"id":3544},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":3542,"freeze":241},"\\frac{S_{\\text{против течения}}}{V_{\\text{против течения}}} + \\frac{S_{\\text{по течению}}}{V_{\\text{по течению}}} = \\frac{S_{\\text{стоячая}}}{V_{\\text{стоячая}}}","$$ \\frac{S_{\\text{против течения}}}{V_{\\text{против течения}}} + \\frac{S_{\\text{по течению}}}{V_{\\text{по течению}}} = \\frac{S_{\\text{стоячая}}}{V_{\\text{стоячая}}} $$","blockMath-QKNNKY7pI",{"__TSPROSE_proseElement":174,"schema":3546,"children":3547,"id":3580},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[3548,3551,3557,3560,3564,3567,3571,3574,3578],{"__TSPROSE_proseElement":174,"schema":3549,"data":3550},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Все расстояния нам известны. Скорость передвижения против течения равна собственной скорости байдарки ",{"__TSPROSE_proseElement":174,"schema":3552,"data":3553,"storageKey":3555,"id":3556},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":3554},"V_б","$ V_б $","inlinerMath-jbyOb7IEx",{"__TSPROSE_proseElement":174,"schema":3558,"data":3559},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," минус скорость течения ",{"__TSPROSE_proseElement":174,"schema":3561,"data":3562,"storageKey":1784,"id":3563},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1783},"inlinerMath-GWJ3eIxvK-4",{"__TSPROSE_proseElement":174,"schema":3565,"data":3566},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},", а скорость по течению равна собственной скорости байдарки ",{"__TSPROSE_proseElement":174,"schema":3568,"data":3569,"storageKey":3555,"id":3570},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":3554},"inlinerMath-jbyOb7IEx-1",{"__TSPROSE_proseElement":174,"schema":3572,"data":3573},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," плюс скорость течения ",{"__TSPROSE_proseElement":174,"schema":3575,"data":3576,"storageKey":1784,"id":3577},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1783},"inlinerMath-GWJ3eIxvK-5",{"__TSPROSE_proseElement":174,"schema":3579,"data":531},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"paragraph-njQURvUOh",{"__TSPROSE_proseElement":174,"schema":3582,"data":3583,"storageKey":3585,"id":3586},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":3584,"freeze":241},"\\frac{15}{V_б - 1} + \\frac{14}{V_б + 1} = \\frac{30}{V_б}","$$ \\frac{15}{V_б - 1} + \\frac{14}{V_б + 1} = \\frac{30}{V_б} $$","blockMath-DBKsVLAAt",{"__TSPROSE_proseElement":174,"schema":3588,"children":3589,"id":3600},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[3590,3593,3597],{"__TSPROSE_proseElement":174,"schema":3591,"data":3592},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Получили уравнение с одной неизвестной, которая, кстати, и является собственной скоростью байдарки ",{"__TSPROSE_proseElement":174,"schema":3594,"data":3595,"storageKey":3555,"id":3596},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":3554},"inlinerMath-jbyOb7IEx-2",{"__TSPROSE_proseElement":174,"schema":3598,"data":3599},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},". 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Они могут покрасить забор за ",{"__TSPROSE_proseElement":174,"schema":4928,"data":4929,"storageKey":614,"id":4930},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":613},"inlinerMath-9rSWhNw25-11",{"__TSPROSE_proseElement":174,"schema":4932,"data":4933},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," часов:","paragraph-MapvIgzFU",{"__TSPROSE_proseElement":174,"schema":4936,"data":4937,"storageKey":4939,"id":4940},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":4938,"freeze":241},"A = (P_1 + P_2) \\cdot 6","$$ A = (P_1 + P_2) \\cdot 6 $$","blockMath-q8bHA4flm",{"__TSPROSE_proseElement":174,"schema":4942,"children":4943,"id":4976},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[4944,4947,4951,4954,4958,4961,4965,4968,4974],{"__TSPROSE_proseElement":174,"schema":4945,"data":4946},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Никаких скоростей у нас нет, поэтому давайте заменим их на выражения через ",{"__TSPROSE_proseElement":174,"schema":4948,"data":4949,"storageKey":1613,"id":4950},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1612},"inlinerMath-eIvv0Olnw-2",{"__TSPROSE_proseElement":174,"schema":4952,"data":4953},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," и время работы. Получить эти выражения можно, если в формуле ",{"__TSPROSE_proseElement":174,"schema":4955,"data":4956,"storageKey":4825,"id":4957},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":4824},"inlinerMath-vdoWz9cQV-2",{"__TSPROSE_proseElement":174,"schema":4959,"data":4960},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," поделить обе части на ",{"__TSPROSE_proseElement":174,"schema":4962,"data":4963,"storageKey":2226,"id":4964},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":2225},"inlinerMath-NdU0V7CGr-2",{"__TSPROSE_proseElement":174,"schema":4966,"data":4967},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," -- тогда мы получим ",{"__TSPROSE_proseElement":174,"schema":4969,"data":4970,"storageKey":4972,"id":4973},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":4971},"P = \\frac{A}{t}","$ P = \\frac{A}{t} $","inlinerMath-7nosJRmwl",{"__TSPROSE_proseElement":174,"schema":4975,"data":531},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"paragraph-WaF5ma1lo",{"__TSPROSE_proseElement":174,"schema":4978,"data":4979,"storageKey":4981,"id":4982},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":4980,"freeze":241},"A = \\left(\\frac{A}{t_1} + \\frac{A}{t_2}\\right) \\cdot 6","$$ A = \\left(\\frac{A}{t_1} + \\frac{A}{t_2}\\right) \\cdot 6 $$","blockMath-PNOGz1iE7",{"__TSPROSE_proseElement":174,"schema":4984,"children":4985,"id":5003},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[4986,4989,4993,4996,5000],{"__TSPROSE_proseElement":174,"schema":4987,"data":4988},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Выносим ",{"__TSPROSE_proseElement":174,"schema":4990,"data":4991,"storageKey":1613,"id":4992},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1612},"inlinerMath-eIvv0Olnw-3",{"__TSPROSE_proseElement":174,"schema":4994,"data":4995},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," за скобки и делим обе части уравнения на ",{"__TSPROSE_proseElement":174,"schema":4997,"data":4998,"storageKey":1613,"id":4999},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1612},"inlinerMath-eIvv0Olnw-4",{"__TSPROSE_proseElement":174,"schema":5001,"data":5002},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},". 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Идеально, ведь никаких других данных задача нам не дает. Ещё нам известно, что первый рабочий может покрасить забор на ",{"__TSPROSE_proseElement":174,"schema":5017,"data":5018,"storageKey":790,"id":5019},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":789},"inlinerMath-MAD3UaxRg-4",{"__TSPROSE_proseElement":174,"schema":5021,"data":5022},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," часов быстрее, чем второй рабочий. На языке математики это можно записать так: ",{"__TSPROSE_proseElement":174,"schema":5024,"data":5025,"storageKey":5027,"id":5028},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":5026},"t_1 = t_2 - 5","$ t_1 = t_2 - 5 $","inlinerMath-zJEWtPi6T",{"__TSPROSE_proseElement":174,"schema":5030,"data":5031},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},". 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Давайте решим его!","paragraph-VzaEfbyoN",{"__TSPROSE_proseElement":174,"schema":5054,"data":5055,"storageKey":5057,"id":5058},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":5056,"freeze":241},"\\frac{(t_2 - 5)t_2}{6} = t_2 + t_2 - 5 \\\\ \\frac{t_2^2 - 5t_2}{6} = 2t_2 - 5 \\\\ t_2^2 - 5t_2 = 12t_2 - 30 \\\\ t_2^2 - 17t_2 + 30 = 0","$$ \\frac{(t_2 - 5)t_2}{6} = t_2 + t_2 - 5 \\\\ \\frac{t_2^2 - 5t_2}{6} = 2t_2 - 5 \\\\ t_2^2 - 5t_2 = 12t_2 - 30 \\\\ t_2^2 - 17t_2 + 30 = 0 $$","blockMath-FT823xSkP",{"__TSPROSE_proseElement":174,"schema":5060,"children":5061,"id":5110},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[5062,5065,5071,5074,5078,5081,5087,5089,5093,5096,5102,5104,5108],{"__TSPROSE_proseElement":174,"schema":5063,"data":5064},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Это квадратное уравнение. Неожиданно, правда? 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Ежедневно первая бригада вспахивала на ",{"__TSPROSE_proseElement":174,"schema":5203,"data":5204,"storageKey":655,"id":5205},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":632},"inlinerMath-mqn34wvB0-19",{"__TSPROSE_proseElement":174,"schema":5207,"data":5208},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," га меньше, чем вторая, но закончила работу на два дня раньше второй. Сколько гектаров земли вспахивала каждая бригада за один рабочий день, если намеченная ежедневная норма в ",{"__TSPROSE_proseElement":174,"schema":5210,"data":5211,"storageKey":1158,"id":5212},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1157},"inlinerMath-pnamxzA0J-1",{"__TSPROSE_proseElement":174,"schema":5214,"data":5215},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," га перевыполнялась обеими бригадами?","paragraph-HZqqvPKyh",{"__TSPROSE_proseElement":174,"schema":5218,"data":5219},{"name":387,"type":240,"linkable":241,"__TSPROSE_schema":174},{"label":5220,"serializedValidator":5221},"Скорость работы бригад",{"type":1169,"ordered":241,"separator":1170,"answers":5222},[5223,5225],{"__ERUDIT_CHECK":174,"name":391,"data":5224},{"expr":4482},{"__ERUDIT_CHECK":174,"name":391,"data":5226},{"expr":5227},"27",{"__TSPROSE_proseElement":174,"schema":5229,"children":5230},{"name":396,"type":240,"linkable":241,"__TSPROSE_schema":174},[5231],{"__TSPROSE_proseElement":174,"schema":5232,"children":5233,"id":5237},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[5234],{"__TSPROSE_proseElement":174,"schema":5235,"data":5236},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"За основу возьмите зависимость между временем работы первой и второй бригады. Вместо времени подставьте выражения через объем работы и скорость работы.","paragraph-8QRn8k1ny",{"__TSPROSE_proseElement":174,"schema":5239,"children":5240},{"name":407,"type":240,"linkable":241,"__TSPROSE_schema":174},[5241],{"__TSPROSE_proseElement":174,"schema":5242,"children":5243,"id":5262},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[5244,5247,5251,5254,5259],{"__TSPROSE_proseElement":174,"schema":5245,"data":5246},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Первая бригада вспахивала по ",{"__TSPROSE_proseElement":174,"schema":5248,"data":5249,"storageKey":4513,"id":5250},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":4482},"inlinerMath-vVUmnH6Sj-2",{"__TSPROSE_proseElement":174,"schema":5252,"data":5253},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," га в день, а вторая бригада вспахивала по ",{"__TSPROSE_proseElement":174,"schema":5255,"data":5256,"storageKey":5257,"id":5258},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":5227},"$ 27 $","inlinerMath-IW9FMi3Jk",{"__TSPROSE_proseElement":174,"schema":5260,"data":5261},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," га в день.","paragraph-hBmIpzkIx",{"__TSPROSE_proseElement":174,"schema":5264,"children":5265},{"name":426,"type":240,"linkable":241,"__TSPROSE_schema":174},[5266,5282,5288,5295,5301,5308,5314,5338,5344,5356,5362,5374,5380],{"__TSPROSE_proseElement":174,"schema":5267,"children":5268,"id":5281},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[5269,5272,5278],{"__TSPROSE_proseElement":174,"schema":5270,"data":5271},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Переводим прямо каждое предложение на язык математики. Сначала нам дали объем работы, которую выполнила первая бригада ",{"__TSPROSE_proseElement":174,"schema":5273,"data":5274,"storageKey":5276,"id":5277},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":5275},"A_1 = 240","$ A_1 = 240 $","inlinerMath-Nvu8JVA6N",{"__TSPROSE_proseElement":174,"schema":5279,"data":5280},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},". Сразу же находим, сколько сделала вторая бригада:","paragraph-GeJjkeR3W",{"__TSPROSE_proseElement":174,"schema":5283,"data":5284,"storageKey":5286,"id":5287},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":5285,"freeze":241},"A_2 = A_1 + \\frac{35}{100} \\cdot A_1 = 240 + \\frac{35}{100} \\cdot 240 = 324","$$ A_2 = A_1 + \\frac{35}{100} \\cdot A_1 = 240 + \\frac{35}{100} \\cdot 240 = 324 $$","blockMath-rdEe3R2Ib",{"__TSPROSE_proseElement":174,"schema":5289,"children":5290,"id":5294},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[5291],{"__TSPROSE_proseElement":174,"schema":5292,"data":5293},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Во втором предложении нам дана зависимость скоростей работы и времени выполнения задачи каждой бригадой. Переводим в равенства:","paragraph-hzeGzDoYV",{"__TSPROSE_proseElement":174,"schema":5296,"data":5297,"storageKey":5299,"id":5300},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":5298,"freeze":241},"P_1 = P_2 - 3 >>{big} t_1 = t_2 - 2","$$ P_1 = P_2 - 3 >>{big} t_1 = t_2 - 2 $$","blockMath-q2VVSWXHM",{"__TSPROSE_proseElement":174,"schema":5302,"children":5303,"id":5307},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[5304],{"__TSPROSE_proseElement":174,"schema":5305,"data":5306},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"За \"основное\" можно принять любое из этих двух равенств. Но так как найти нам требуется скорость, то за основу возьмём второе равенство и подставим в него выражения для времени через объем работы и скорость:","paragraph-VXY2bNUAD",{"__TSPROSE_proseElement":174,"schema":5309,"data":5310,"storageKey":5312,"id":5313},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":5311,"freeze":241},"\\frac{A_1}{P_1} = \\frac{A_2}{P_2} - 2","$$ \\frac{A_1}{P_1} = \\frac{A_2}{P_2} - 2 $$","blockMath-5HSu53Mo9",{"__TSPROSE_proseElement":174,"schema":5315,"children":5316,"id":5337},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[5317,5320,5326,5328,5334],{"__TSPROSE_proseElement":174,"schema":5318,"data":5319},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Теперь подставляем в это уравнение выражение для ",{"__TSPROSE_proseElement":174,"schema":5321,"data":5322,"storageKey":5324,"id":5325},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":5323},"P_1","$ P_1 $","inlinerMath-hvGY9z42D",{"__TSPROSE_proseElement":174,"schema":5327,"data":1944},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":5329,"data":5330,"storageKey":5332,"id":5333},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":5331},"P_2","$ P_2 $","inlinerMath-UcV5Mm3dY",{"__TSPROSE_proseElement":174,"schema":5335,"data":5336},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," из первого равенства, а также численные значения объемов работы:","paragraph-sUvxS47pe",{"__TSPROSE_proseElement":174,"schema":5339,"data":5340,"storageKey":5342,"id":5343},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":5341,"freeze":241},"\\frac{240}{P_2 - 3} = \\frac{324}{P_2} - 2 \\\\ 240P_2 = 324(P_2 - 3) - 2P_2(P_2 - 3) \\\\ 120P_2 = 162(P_2 - 3) - P_2(P_2 - 3) \\\\ 120P_2 = 162P_2 - 486 - P_2^2 + 3P_2 \\\\ P_2^2 - 45P_2 + 486 = 0","$$ \\frac{240}{P_2 - 3} = \\frac{324}{P_2} - 2 \\\\ 240P_2 = 324(P_2 - 3) - 2P_2(P_2 - 3) \\\\ 120P_2 = 162(P_2 - 3) - P_2(P_2 - 3) \\\\ 120P_2 = 162P_2 - 486 - P_2^2 + 3P_2 \\\\ P_2^2 - 45P_2 + 486 = 0 $$","blockMath-EYBdS4D5G",{"__TSPROSE_proseElement":174,"schema":5345,"children":5346,"id":5355},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[5347,5349,5353],{"__TSPROSE_proseElement":174,"schema":5348,"data":3612},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":5350,"data":5351,"storageKey":2419,"id":5352},{"name":488,"type":250,"linkable":174,"__TSPROSE_schema":174},{"label":2418},"dependencyInliner-WuvFqSz1Q-3",{"__TSPROSE_proseElement":174,"schema":5354,"data":449},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"paragraph-qcLLGIb1a-1",{"__TSPROSE_proseElement":174,"schema":5357,"data":5358,"storageKey":5360,"id":5361},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":5359,"freeze":241},"D = (-45)^2 - 4 \\cdot 1 \\cdot 486 = 2025 - 1944 = 81","$$ D = (-45)^2 - 4 \\cdot 1 \\cdot 486 = 2025 - 1944 = 81 $$","blockMath-QYHhsA97U",{"__TSPROSE_proseElement":174,"schema":5363,"children":5364,"id":5373},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[5365,5367,5371],{"__TSPROSE_proseElement":174,"schema":5366,"data":3631},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":5368,"data":5369,"storageKey":2440,"id":5370},{"name":488,"type":250,"linkable":174,"__TSPROSE_schema":174},{"label":3635},"dependencyInliner-l1AilHmCR-1",{"__TSPROSE_proseElement":174,"schema":5372,"data":449},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"paragraph-7vMYXXoO7-1",{"__TSPROSE_proseElement":174,"schema":5375,"data":5376,"storageKey":5378,"id":5379},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":5377,"freeze":241},"P_2 = \\frac{-(-45) \\pm \\sqrt{81}}{2 \\cdot 1} = \\frac{45 \\pm 9}{2}","$$ P_2 = \\frac{-(-45) \\pm \\sqrt{81}}{2 \\cdot 1} = \\frac{45 \\pm 9}{2} $$","blockMath-KsflWUvv2",{"__TSPROSE_proseElement":174,"schema":5381,"children":5382,"id":5432},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[5383,5385,5391,5393,5399,5402,5406,5409,5413,5416,5420,5423,5429],{"__TSPROSE_proseElement":174,"schema":5384,"data":2053},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":5386,"data":5387,"storageKey":5389,"id":5390},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":5388},"P_2 = 27","$ P_2 = 27 $","inlinerMath-e2712Cpbw",{"__TSPROSE_proseElement":174,"schema":5392,"data":522},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":5394,"data":5395,"storageKey":5397,"id":5398},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":5396},"P_2 = 18","$ P_2 = 18 $","inlinerMath-kUpHO4Dre",{"__TSPROSE_proseElement":174,"schema":5400,"data":5401},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},". 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Так что остается только вариант ",{"__TSPROSE_proseElement":174,"schema":5417,"data":5418,"storageKey":5389,"id":5419},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":5388},"inlinerMath-e2712Cpbw-1",{"__TSPROSE_proseElement":174,"schema":5421,"data":5422},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," гектаров в день, а значит ",{"__TSPROSE_proseElement":174,"schema":5424,"data":5425,"storageKey":5427,"id":5428},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":5426},"P_1 = 27 - 3 = 24","$ P_1 = 27 - 3 = 24 $","inlinerMath-cA19QOTTg",{"__TSPROSE_proseElement":174,"schema":5430,"data":5431},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," гектара в день.","paragraph-YYdp84nXc","vspakhivanie-polya",{"__TSPROSE_proseElement":174,"schema":5435,"data":5436,"children":5440,"id":5780},{"name":332,"type":240,"linkable":174,"__TSPROSE_schema":174},{"info":5437},{"title":5438,"level":1412,"attributes":5439},"3 крана -- 1 бак",[],[5441,5474,5487,5497,5530],{"__TSPROSE_proseElement":174,"schema":5442,"children":5443},{"name":341,"type":240,"linkable":241,"__TSPROSE_schema":174},[5444],{"__TSPROSE_proseElement":174,"schema":5445,"children":5446,"id":5473},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[5447,5450,5454,5457,5463,5466,5470],{"__TSPROSE_proseElement":174,"schema":5448,"data":5449},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Некоторая жидкость поступает в бак через три входных крана. Если открыть все краны одновременно, то бак наполнится за ",{"__TSPROSE_proseElement":174,"schema":5451,"data":5452,"storageKey":614,"id":5453},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":613},"inlinerMath-9rSWhNw25-12",{"__TSPROSE_proseElement":174,"schema":5455,"data":5456},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," минут. Если же наполнять бак только через второй кран, то на это потребуется ",{"__TSPROSE_proseElement":174,"schema":5458,"data":5459,"storageKey":5461,"id":5462},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":5460},"3\u002F4","$ 3\u002F4 $","inlinerMath-7pQOp7GKd",{"__TSPROSE_proseElement":174,"schema":5464,"data":5465},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," того времени, за которое может наполниться бак только через один первый кран. Через один третий кран этот бак наполняется на ",{"__TSPROSE_proseElement":174,"schema":5467,"data":5468,"storageKey":1077,"id":5469},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1076},"inlinerMath-8Y3Z60Vbg-9",{"__TSPROSE_proseElement":174,"schema":5471,"data":5472},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," минут дольше, чем через один второй кран. На какое время надо открывать каждый кран в отдельности для наполнения бака?","paragraph-prSW3XGUt",{"__TSPROSE_proseElement":174,"schema":5475,"data":5476},{"name":387,"type":240,"linkable":241,"__TSPROSE_schema":174},{"label":5477,"serializedValidator":5478},"Время наполнения бака каждым краном",{"type":1169,"ordered":241,"separator":1170,"answers":5479},[5480,5483,5485],{"__ERUDIT_CHECK":174,"name":391,"data":5481},{"expr":5482},"56\u002F3",{"__ERUDIT_CHECK":174,"name":391,"data":5484},{"expr":3403},{"__ERUDIT_CHECK":174,"name":391,"data":5486},{"expr":4482},{"__TSPROSE_proseElement":174,"schema":5488,"children":5489},{"name":396,"type":240,"linkable":241,"__TSPROSE_schema":174},[5490],{"__TSPROSE_proseElement":174,"schema":5491,"children":5492,"id":5496},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[5493],{"__TSPROSE_proseElement":174,"schema":5494,"data":5495},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"За основу возьмите формулу работы (заполнение всего бака) для ситуации, когда все краны открыты одновременно. В сумме скоростей замените каждую из скоростей на выражение через объем работы и время работы.","paragraph-uwlPK6rfm",{"__TSPROSE_proseElement":174,"schema":5498,"children":5499},{"name":407,"type":240,"linkable":241,"__TSPROSE_schema":174},[5500],{"__TSPROSE_proseElement":174,"schema":5501,"children":5502,"id":5529},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[5503,5506,5512,5515,5519,5522,5526],{"__TSPROSE_proseElement":174,"schema":5504,"data":5505},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Первый кран наполняет бак за ",{"__TSPROSE_proseElement":174,"schema":5507,"data":5508,"storageKey":5510,"id":5511},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":5509},"\\frac{56}{3}","$ \\frac{56}{3} $","inlinerMath-3LOq1XIcj",{"__TSPROSE_proseElement":174,"schema":5513,"data":5514},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," минут, второй кран наполняет бак за ",{"__TSPROSE_proseElement":174,"schema":5516,"data":5517,"storageKey":3404,"id":5518},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":3403},"inlinerMath-EG1ZwIV76-1",{"__TSPROSE_proseElement":174,"schema":5520,"data":5521},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," минут, а третий кран наполняет бак за ",{"__TSPROSE_proseElement":174,"schema":5523,"data":5524,"storageKey":4513,"id":5525},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":4482},"inlinerMath-vVUmnH6Sj-3",{"__TSPROSE_proseElement":174,"schema":5527,"data":5528},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," минуты.","paragraph-cqxXScFbo",{"__TSPROSE_proseElement":174,"schema":5531,"children":5532},{"name":426,"type":240,"linkable":241,"__TSPROSE_schema":174},[5533,5540,5546,5553,5559,5566,5572,5593,5599,5619,5625,5648,5654,5670,5676,5688,5694,5706,5712,5749,5755],{"__TSPROSE_proseElement":174,"schema":5534,"children":5535,"id":5539},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[5536],{"__TSPROSE_proseElement":174,"schema":5537,"data":5538},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Выпишем формулу по наполнению всего бака сразу через все краны:","paragraph-htmwXZ8Op",{"__TSPROSE_proseElement":174,"schema":5541,"data":5542,"storageKey":5544,"id":5545},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":5543,"freeze":241},"A = (P_1 + P_2 + P_3) \\cdot 6","$$ A = (P_1 + P_2 + P_3) \\cdot 6 $$","blockMath-HdYKLFag5",{"__TSPROSE_proseElement":174,"schema":5547,"children":5548,"id":5552},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[5549],{"__TSPROSE_proseElement":174,"schema":5550,"data":5551},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Теперь выпишем все зависимости по времени:","paragraph-bz48FzE8V",{"__TSPROSE_proseElement":174,"schema":5554,"data":5555,"storageKey":5557,"id":5558},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":5556,"freeze":241},"t_2 = \\frac{3}{4}t_1 >>{big} t_3 = t_2 + 10","$$ t_2 = \\frac{3}{4}t_1 >>{big} t_3 = t_2 + 10 $$","blockMath-U2hHGi5fM",{"__TSPROSE_proseElement":174,"schema":5560,"children":5561,"id":5565},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[5562],{"__TSPROSE_proseElement":174,"schema":5563,"data":5564},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Подставим в первую формулу вместо скоростей выражения через объем работы и время, потому что именно его (время) нам и нужно найти:","paragraph-1PqkHzYw9",{"__TSPROSE_proseElement":174,"schema":5567,"data":5568,"storageKey":5570,"id":5571},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":5569,"freeze":241},"A = \\left(\\frac{A}{t_1} + \\frac{A}{t_2} + \\frac{A}{t_3}\\right) \\cdot 6","$$ A = \\left(\\frac{A}{t_1} + \\frac{A}{t_2} + \\frac{A}{t_3}\\right) \\cdot 6 $$","blockMath-HZKgrfYU1",{"__TSPROSE_proseElement":174,"schema":5573,"children":5574,"id":5592},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[5575,5578,5582,5585,5589],{"__TSPROSE_proseElement":174,"schema":5576,"data":5577},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Выносим за скобки ",{"__TSPROSE_proseElement":174,"schema":5579,"data":5580,"storageKey":1613,"id":5581},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1612},"inlinerMath-eIvv0Olnw-5",{"__TSPROSE_proseElement":174,"schema":5583,"data":5584},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," справа и делим обе части на ",{"__TSPROSE_proseElement":174,"schema":5586,"data":5587,"storageKey":1613,"id":5588},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1612},"inlinerMath-eIvv0Olnw-6",{"__TSPROSE_proseElement":174,"schema":5590,"data":5591},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},". 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Время не может быть отрицательным, поэтому ",{"__TSPROSE_proseElement":174,"schema":5735,"data":5736,"storageKey":5721,"id":5737},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":5720},"inlinerMath-qQrNM4bji-1",{"__TSPROSE_proseElement":174,"schema":5739,"data":5740},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," минут. 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В результате получаются смеси (газов), растворы (жидкостей) и сплавы (твердых веществ). Это всё разные названия для одного и того же!","paragraph-EgFxsqXPG",{"__TSPROSE_proseElement":174,"schema":5794,"children":5795,"id":5836},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[5796,5799,5807,5810,5818,5821,5825,5828,5834],{"__TSPROSE_proseElement":174,"schema":5797,"data":5798},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Для всех этих смесей, растворов и сплавов очень важно понимать, ",{"__TSPROSE_proseElement":174,"schema":5800,"data":5801,"children":5802,"id":5806},{"name":269,"type":250,"linkable":174,"__TSPROSE_schema":174},{"type":271},[5803],{"__TSPROSE_proseElement":174,"schema":5804,"data":5805},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"какую долю","emphasis-8S3jCMfNW",{"__TSPROSE_proseElement":174,"schema":5808,"data":5809},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," от общей массы\u002Fобъема смеси\u002Fраствора\u002Fсплава составляет интересующее нас вещество. 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Её можно посчитать через формулу \"концентрации\" меди в этом слитке:","paragraph-ee1zqoJj6",{"__TSPROSE_proseElement":174,"schema":6252,"data":6253,"storageKey":6255,"id":6256},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":6254,"freeze":241},"0.36 = \\frac{6 + 12}{M} \\implies M = \\frac{18}{0.36} = \\frac{18 \\cdot 100}{36} = 50","$$ 0.36 = \\frac{6 + 12}{M} \\implies M = \\frac{18}{0.36} = \\frac{18 \\cdot 100}{36} = 50 $$","blockMath-7BaMXVaEn",{"__TSPROSE_proseElement":174,"schema":6258,"children":6259,"id":6263},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[6260],{"__TSPROSE_proseElement":174,"schema":6261,"data":6262},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Выпишем в виде формулы зависимость концентрации меди в исходных сплавах:","paragraph-tX6CNDyOx",{"__TSPROSE_proseElement":174,"schema":6265,"data":6266,"storageKey":6268,"id":6269},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":6267,"freeze":241},"c_1 = c_2 - 0.4","$$ c_1 = c_2 - 0.4 $$","blockMath-6WDVoyNdw",{"__TSPROSE_proseElement":174,"schema":6271,"children":6272,"id":6276},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[6273],{"__TSPROSE_proseElement":174,"schema":6274,"data":6275},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Пока что с этой формулой мы ничего сделать не можем. Подставлять вместо концентрации какие-то выражения не хочется, ведь нам концентрацию и нужно найти, а не чем-то её заменить! Пойдём другим путём. Вспоминаем, что слиток мы получили из двух кусков сплавов. Запишем это на языке математики:","paragraph-Ouf3YS7fE",{"__TSPROSE_proseElement":174,"schema":6278,"data":6279,"storageKey":6281,"id":6282},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":6280,"freeze":241},"m_1 + m_2 = 50","$$ m_1 + m_2 = 50 $$","blockMath-3rUQ9jK6J",{"__TSPROSE_proseElement":174,"schema":6284,"children":6285,"id":6289},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[6286],{"__TSPROSE_proseElement":174,"schema":6287,"data":6288},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"А вот масса куска уже напрямую связана с концентрацией меди в этом куске:","paragraph-oADzom79s",{"__TSPROSE_proseElement":174,"schema":6291,"data":6292,"storageKey":6294,"id":6295},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":6293,"freeze":241},"c_1 = \\frac{6}{m_1} \\implies m_1 = \\frac{6}{c_1} >>{big} c_2 = \\frac{12}{m_2} \\implies m_2 = \\frac{12}{c_2}","$$ c_1 = \\frac{6}{m_1} \\implies m_1 = \\frac{6}{c_1} >>{big} c_2 = \\frac{12}{m_2} \\implies m_2 = \\frac{12}{c_2} $$","blockMath-dX20hBUQv",{"__TSPROSE_proseElement":174,"schema":6297,"children":6298,"id":6302},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[6299],{"__TSPROSE_proseElement":174,"schema":6300,"data":6301},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Подставляем вместо масс кусков дроби с концентрацией:","paragraph-vhswP9sAp",{"__TSPROSE_proseElement":174,"schema":6304,"data":6305,"storageKey":6307,"id":6308},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":6306,"freeze":241},"\\frac{6}{c_1} + \\frac{12}{c_2} = 50","$$ \\frac{6}{c_1} + \\frac{12}{c_2} = 50 $$","blockMath-Qs5jyMxjm",{"__TSPROSE_proseElement":174,"schema":6310,"children":6311,"id":6315},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[6312],{"__TSPROSE_proseElement":174,"schema":6313,"data":6314},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Заменяем первую концентрацию на её выражение через вторую:","paragraph-T9kgbh6EJ",{"__TSPROSE_proseElement":174,"schema":6317,"data":6318,"storageKey":6320,"id":6321},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":6319,"freeze":241},"\\frac{6}{c_2 - 0.4} + \\frac{12}{c_2} = 50","$$ \\frac{6}{c_2 - 0.4} + \\frac{12}{c_2} = 50 $$","blockMath-eQZPiQDRQ",{"__TSPROSE_proseElement":174,"schema":6323,"children":6324,"id":6328},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[6325],{"__TSPROSE_proseElement":174,"schema":6326,"data":6327},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Отлично, получили уравнение с одной неизвестной! Решаем его:","paragraph-YmFpOaFDY",{"__TSPROSE_proseElement":174,"schema":6330,"data":6331,"storageKey":6333,"id":6334},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":6332,"freeze":241},"\\frac{6}{c_2 - \\frac{4}{10}} + \\frac{12}{c_2} = 50 \\\\ \\frac{3}{c_2 - \\frac{4}{10}} + \\frac{6}{c_2} = 25 \\\\ 3c_2 + 6\\left(c_2 - \\frac{4}{10}\\right) = 25c_2\\left(c_2 - \\frac{4}{10}\\right) \\\\ 3c_2 + 6c_2 - \\frac{24}{10} = 25c_2^2 - 10c_2 \\\\ 0 = 25c_2^2 - 19c_2 + \\frac{12}{5}","$$ \\frac{6}{c_2 - \\frac{4}{10}} + \\frac{12}{c_2} = 50 \\\\ \\frac{3}{c_2 - \\frac{4}{10}} + \\frac{6}{c_2} = 25 \\\\ 3c_2 + 6\\left(c_2 - \\frac{4}{10}\\right) = 25c_2\\left(c_2 - \\frac{4}{10}\\right) \\\\ 3c_2 + 6c_2 - \\frac{24}{10} = 25c_2^2 - 10c_2 \\\\ 0 = 25c_2^2 - 19c_2 + \\frac{12}{5} $$","blockMath-VhSeH38U5",{"__TSPROSE_proseElement":174,"schema":6336,"children":6337,"id":6346},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[6338,6340,6344],{"__TSPROSE_proseElement":174,"schema":6339,"data":3612},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":6341,"data":6342,"storageKey":2419,"id":6343},{"name":488,"type":250,"linkable":174,"__TSPROSE_schema":174},{"label":2418},"dependencyInliner-WuvFqSz1Q-5",{"__TSPROSE_proseElement":174,"schema":6345,"data":449},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"paragraph-qcLLGIb1a-3",{"__TSPROSE_proseElement":174,"schema":6348,"data":6349,"storageKey":6351,"id":6352},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":6350,"freeze":241},"D = (-19)^2 - 4 \\cdot 25 \\cdot \\frac{12}{5} = 361 - 240 = 121","$$ D = (-19)^2 - 4 \\cdot 25 \\cdot \\frac{12}{5} = 361 - 240 = 121 $$","blockMath-B7oYzteU1",{"__TSPROSE_proseElement":174,"schema":6354,"children":6355,"id":6364},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[6356,6358,6362],{"__TSPROSE_proseElement":174,"schema":6357,"data":3631},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":6359,"data":6360,"storageKey":2440,"id":6361},{"name":488,"type":250,"linkable":174,"__TSPROSE_schema":174},{"label":3635},"dependencyInliner-l1AilHmCR-3",{"__TSPROSE_proseElement":174,"schema":6363,"data":449},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"paragraph-7vMYXXoO7-3",{"__TSPROSE_proseElement":174,"schema":6366,"data":6367,"storageKey":6369,"id":6370},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":6368,"freeze":241},"c_2 = \\frac{-(-19) \\pm \\sqrt{121}}{2 \\cdot 25} = \\frac{19 \\pm 11}{50}","$$ c_2 = \\frac{-(-19) \\pm \\sqrt{121}}{2 \\cdot 25} = \\frac{19 \\pm 11}{50} $$","blockMath-QOpJOwP7M",{"__TSPROSE_proseElement":174,"schema":6372,"children":6373,"id":6431},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[6374,6376,6382,6384,6390,6393,6397,6400,6406,6409,6413,6416,6422,6425,6429],{"__TSPROSE_proseElement":174,"schema":6375,"data":2053},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":6377,"data":6378,"storageKey":6380,"id":6381},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":6379},"c_2 = 0.6","$ c_2 = 0.6 $","inlinerMath-Uy914ip7y",{"__TSPROSE_proseElement":174,"schema":6383,"data":522},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":6385,"data":6386,"storageKey":6388,"id":6389},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":6387},"c_2 = 0.16","$ c_2 = 0.16 $","inlinerMath-vHwshtYGl",{"__TSPROSE_proseElement":174,"schema":6391,"data":6392},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},". 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Для начала разделим обе части уравнения на ",{"__TSPROSE_proseElement":174,"schema":7767,"data":7768,"storageKey":418,"id":7769},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":393},"inlinerMath-z9Yar9waf-27",{"__TSPROSE_proseElement":174,"schema":7771,"data":766},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"paragraph-oc3F48fBq",{"__TSPROSE_proseElement":174,"schema":7774,"data":7775,"storageKey":7777,"id":7778},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":7776,"freeze":241},"b^2 - 23b + 120 = 0","$$ b^2 - 23b + 120 = 0 $$","blockMath-esvfs4b7Q",{"__TSPROSE_proseElement":174,"schema":7780,"children":7781,"id":7798},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[7782,7785,7789,7792,7796],{"__TSPROSE_proseElement":174,"schema":7783,"data":7784},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Находим ",{"__TSPROSE_proseElement":174,"schema":7786,"data":7787,"storageKey":2419,"id":7788},{"name":488,"type":250,"linkable":174,"__TSPROSE_schema":174},{"label":2418},"dependencyInliner-WuvFqSz1Q-7",{"__TSPROSE_proseElement":174,"schema":7790,"data":7791},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," и решаем квадратное уравнение по ",{"__TSPROSE_proseElement":174,"schema":7793,"data":7794,"storageKey":2440,"id":7795},{"name":488,"type":250,"linkable":174,"__TSPROSE_schema":174},{"label":3635},"dependencyInliner-l1AilHmCR-5",{"__TSPROSE_proseElement":174,"schema":7797,"data":449},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"paragraph-DvWWRxmWF",{"__TSPROSE_proseElement":174,"schema":7800,"data":7801,"storageKey":7803,"id":7804},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":7802,"freeze":241},"D = (-23)^2 - 4 \\cdot 1 \\cdot 120 = 529 - 480 = 49 \\\\ b = \\frac{-(-23) \\pm \\sqrt{49}}{2 \\cdot 1} = \\frac{23 \\pm 7}{2}","$$ D = (-23)^2 - 4 \\cdot 1 \\cdot 120 = 529 - 480 = 49 \\\\ b = \\frac{-(-23) \\pm \\sqrt{49}}{2 \\cdot 1} = \\frac{23 \\pm 7}{2} $$","blockMath-WCt5uozL4",{"__TSPROSE_proseElement":174,"schema":7806,"children":7807,"id":7875},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[7808,7810,7816,7818,7824,7827,7831,7834,7838,7840,7846,7848,7852,7854,7860,7863,7867,7869,7873],{"__TSPROSE_proseElement":174,"schema":7809,"data":2053},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":7811,"data":7812,"storageKey":7814,"id":7815},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":7813},"b = 15","$ b = 15 $","inlinerMath-2mflkohXG",{"__TSPROSE_proseElement":174,"schema":7817,"data":522},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":7819,"data":7820,"storageKey":7822,"id":7823},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":7821},"b = 8","$ b = 8 $","inlinerMath-8ClzqcZVa",{"__TSPROSE_proseElement":174,"schema":7825,"data":7826},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},". Вроде оба выглядят нормально, но посмотрим, что будет с ",{"__TSPROSE_proseElement":174,"schema":7828,"data":7829,"storageKey":1504,"id":7830},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1503},"inlinerMath-YqDfaaRXA-7",{"__TSPROSE_proseElement":174,"schema":7832,"data":7833},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," в каждом случае. Если ",{"__TSPROSE_proseElement":174,"schema":7835,"data":7836,"storageKey":7814,"id":7837},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":7813},"inlinerMath-2mflkohXG-1",{"__TSPROSE_proseElement":174,"schema":7839,"data":6399},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":7841,"data":7842,"storageKey":7844,"id":7845},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":7843},"a = 8","$ a = 8 $","inlinerMath-zPiKVzSk7",{"__TSPROSE_proseElement":174,"schema":7847,"data":6392},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":7849,"data":7850,"storageKey":7822,"id":7851},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":7821},"inlinerMath-8ClzqcZVa-1",{"__TSPROSE_proseElement":174,"schema":7853,"data":6399},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":7855,"data":7856,"storageKey":7858,"id":7859},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":7857},"a = 15","$ a = 15 $","inlinerMath-Bn4Q7hw4R",{"__TSPROSE_proseElement":174,"schema":7861,"data":7862},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},". Короче, оба корня дают нам одни и те же стороны прямоугольника, просто под разными буквенными обозначениями. Поэтому стороны прямоугольника равны ",{"__TSPROSE_proseElement":174,"schema":7864,"data":7865,"storageKey":1766,"id":7866},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1765},"inlinerMath-1xC295UiR-6",{"__TSPROSE_proseElement":174,"schema":7868,"data":522},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":7870,"data":7871,"storageKey":806,"id":7872},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":805},"inlinerMath-fBsW71uIl-11",{"__TSPROSE_proseElement":174,"schema":7874,"data":6946},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"paragraph-zN0QTxceb","storony-pryamougolnika",{"__TSPROSE_proseElement":174,"schema":7878,"data":7879,"children":7883,"id":8090},{"name":332,"type":240,"linkable":174,"__TSPROSE_schema":174},{"info":7880},{"title":7881,"level":336,"attributes":7882},"Стороны треугольника",[],[7884,7910,7919,7929,7951],{"__TSPROSE_proseElement":174,"schema":7885,"children":7886},{"name":341,"type":240,"linkable":241,"__TSPROSE_schema":174},[7887],{"__TSPROSE_proseElement":174,"schema":7888,"children":7889,"id":7909},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[7890,7893,7897,7900,7906],{"__TSPROSE_proseElement":174,"schema":7891,"data":7892},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Гипотенуза прямоугольного треугольника равна ",{"__TSPROSE_proseElement":174,"schema":7894,"data":7895,"storageKey":790,"id":7896},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":789},"inlinerMath-MAD3UaxRg-5",{"__TSPROSE_proseElement":174,"schema":7898,"data":7899},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},". Если утроить меньшую сторону и увеличить в десять раз большую сторону, то гипотенуза такого увеличенного треугольника будет равна ",{"__TSPROSE_proseElement":174,"schema":7901,"data":7902,"storageKey":7904,"id":7905},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":7903},"41","$ 41 $","inlinerMath-wulfooLHq",{"__TSPROSE_proseElement":174,"schema":7907,"data":7908},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},". Найдите длины катетов исходного треугольника.","paragraph-1JYMBhU0J",{"__TSPROSE_proseElement":174,"schema":7911,"data":7912},{"name":387,"type":240,"linkable":241,"__TSPROSE_schema":174},{"label":7881,"serializedValidator":7913},{"type":1169,"ordered":241,"separator":1170,"answers":7914},[7915,7917],{"__ERUDIT_CHECK":174,"name":391,"data":7916},{"expr":632},{"__ERUDIT_CHECK":174,"name":391,"data":7918},{"expr":604},{"__TSPROSE_proseElement":174,"schema":7920,"children":7921},{"name":396,"type":240,"linkable":241,"__TSPROSE_schema":174},[7922],{"__TSPROSE_proseElement":174,"schema":7923,"children":7924,"id":7928},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[7925],{"__TSPROSE_proseElement":174,"schema":7926,"data":7927},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Воспользуйтесь теоремой Пифагора, чтобы связать стороны треугольника с его гипотенузой.","paragraph-TN2snp8IZ",{"__TSPROSE_proseElement":174,"schema":7930,"children":7931},{"name":407,"type":240,"linkable":241,"__TSPROSE_schema":174},[7932],{"__TSPROSE_proseElement":174,"schema":7933,"children":7934,"id":7950},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[7935,7938,7942,7944,7948],{"__TSPROSE_proseElement":174,"schema":7936,"data":7937},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Катеты треугольника равны ",{"__TSPROSE_proseElement":174,"schema":7939,"data":7940,"storageKey":655,"id":7941},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":632},"inlinerMath-mqn34wvB0-20",{"__TSPROSE_proseElement":174,"schema":7943,"data":522},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":7945,"data":7946,"storageKey":605,"id":7947},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":604},"inlinerMath-ZXyaHjumL-8",{"__TSPROSE_proseElement":174,"schema":7949,"data":531},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"paragraph-l1Y2DN2o9",{"__TSPROSE_proseElement":174,"schema":7952,"children":7953},{"name":426,"type":240,"linkable":241,"__TSPROSE_schema":174},[7954,7961,7967,7987,7993,8000,8006,8045,8051,8071,8077],{"__TSPROSE_proseElement":174,"schema":7955,"children":7956,"id":7960},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[7957],{"__TSPROSE_proseElement":174,"schema":7958,"data":7959},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Катеты и гипотенуза прямоугольного треугольника связаны теоремой Пифагора:","paragraph-Wg7jUo3dI",{"__TSPROSE_proseElement":174,"schema":7962,"data":7963,"storageKey":7965,"id":7966},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":7964,"freeze":241},"a^2 + b^2 = c^2","$$ a^2 + b^2 = c^2 $$","blockMath-91oa9Ng8P",{"__TSPROSE_proseElement":174,"schema":7968,"children":7969,"id":7986},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[7970,7973,7977,7979,7983],{"__TSPROSE_proseElement":174,"schema":7971,"data":7972},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Обозначим катеты исходного треугольника за ",{"__TSPROSE_proseElement":174,"schema":7974,"data":7975,"storageKey":1504,"id":7976},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1503},"inlinerMath-YqDfaaRXA-8",{"__TSPROSE_proseElement":174,"schema":7978,"data":522},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":7980,"data":7981,"storageKey":7325,"id":7982},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":7324},"inlinerMath-LTS8nNQCZ-5",{"__TSPROSE_proseElement":174,"schema":7984,"data":7985},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},". 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Из одной формулы выразите какую-нибудь сторону и вставьте это выражение в другую формулу.","paragraph-1pOg9RM2h",{"__TSPROSE_proseElement":174,"schema":8143,"children":8144},{"name":407,"type":240,"linkable":241,"__TSPROSE_schema":174},[8145],{"__TSPROSE_proseElement":174,"schema":8146,"children":8147,"id":8162},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[8148,8150,8154,8156,8160],{"__TSPROSE_proseElement":174,"schema":8149,"data":7629},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":8151,"data":8152,"storageKey":790,"id":8153},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":789},"inlinerMath-MAD3UaxRg-6",{"__TSPROSE_proseElement":174,"schema":8155,"data":522},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":8157,"data":8158,"storageKey":1077,"id":8159},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1076},"inlinerMath-8Y3Z60Vbg-12",{"__TSPROSE_proseElement":174,"schema":8161,"data":531},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"paragraph-4gfH6459u",{"__TSPROSE_proseElement":174,"schema":8164,"children":8165},{"name":426,"type":240,"linkable":241,"__TSPROSE_schema":174},[8166,8186,8192,8228,8234,8272,8278,8307,8313],{"__TSPROSE_proseElement":174,"schema":8167,"children":8168,"id":8185},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[8169,8172,8176,8178,8182],{"__TSPROSE_proseElement":174,"schema":8170,"data":8171},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Обозначим стороны прямоугольника за ",{"__TSPROSE_proseElement":174,"schema":8173,"data":8174,"storageKey":1504,"id":8175},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1503},"inlinerMath-YqDfaaRXA-10",{"__TSPROSE_proseElement":174,"schema":8177,"data":522},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":8179,"data":8180,"storageKey":7325,"id":8181},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":7324},"inlinerMath-LTS8nNQCZ-6",{"__TSPROSE_proseElement":174,"schema":8183,"data":8184},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},". 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Найдём его корни по ",{"__TSPROSE_proseElement":174,"schema":8649,"data":8650,"storageKey":1343,"id":8651},{"name":488,"type":250,"linkable":174,"__TSPROSE_schema":174},{"label":6061},"dependencyInliner-UuVLCSlbA-1",{"__TSPROSE_proseElement":174,"schema":8653,"data":8654},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," для случая, когда коэффициент ",{"__TSPROSE_proseElement":174,"schema":8656,"data":8657,"storageKey":1334,"id":8658},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1333},"inlinerMath-RvASDBkSH-3",{"__TSPROSE_proseElement":174,"schema":8660,"data":8661},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," чётный:","paragraph-rLQKvLYYl",{"__TSPROSE_proseElement":174,"schema":8664,"data":8665,"storageKey":8667,"id":8668},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":8666,"freeze":241},"b = \\frac{-(-8) \\pm \\sqrt{(-8)^2 - 9 \\cdot (-448)}}{9} = \\frac{8 \\pm \\sqrt{64 + 4032}}{9} = \\frac{8 \\pm \\sqrt{4096}}{9} = \\frac{8 \\pm 64}{9}","$$ b = \\frac{-(-8) \\pm \\sqrt{(-8)^2 - 9 \\cdot (-448)}}{9} = \\frac{8 \\pm \\sqrt{64 + 4032}}{9} = \\frac{8 \\pm \\sqrt{4096}}{9} = \\frac{8 \\pm 64}{9} $$","blockMath-N5uKphwan",{"__TSPROSE_proseElement":174,"schema":8670,"children":8671,"id":8709},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[8672,8674,8678,8680,8686,8689,8693,8696,8700,8703,8707],{"__TSPROSE_proseElement":174,"schema":8673,"data":2053},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":8675,"data":8676,"storageKey":7822,"id":8677},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":7821},"inlinerMath-8ClzqcZVa-2",{"__TSPROSE_proseElement":174,"schema":8679,"data":522},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":8681,"data":8682,"storageKey":8684,"id":8685},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":8683},"b = -\\frac{56}{9}","$ b = -\\frac{56}{9} $","inlinerMath-J7q8D1AKT",{"__TSPROSE_proseElement":174,"schema":8687,"data":8688},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},". Отрицательное значение длины стороны квадрата не имеет физического смысла, поэтому сторона второго квадрата равна ",{"__TSPROSE_proseElement":174,"schema":8690,"data":8691,"storageKey":806,"id":8692},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":805},"inlinerMath-fBsW71uIl-13",{"__TSPROSE_proseElement":174,"schema":8694,"data":8695},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},". Находим сторону другого квадрата, подставив ",{"__TSPROSE_proseElement":174,"schema":8697,"data":8698,"storageKey":7822,"id":8699},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":7821},"inlinerMath-8ClzqcZVa-3",{"__TSPROSE_proseElement":174,"schema":8701,"data":8702},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174}," в выражение для ",{"__TSPROSE_proseElement":174,"schema":8704,"data":8705,"storageKey":1504,"id":8706},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1503},"inlinerMath-YqDfaaRXA-18",{"__TSPROSE_proseElement":174,"schema":8708,"data":449},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"paragraph-tl2snV2Ev",{"__TSPROSE_proseElement":174,"schema":8711,"data":8712,"storageKey":8714,"id":8715},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":8713,"freeze":241},"a = \\frac{5}{4}b = \\frac{5}{4} \\cdot 8 = 10","$$ a = \\frac{5}{4}b = \\frac{5}{4} \\cdot 8 = 10 $$","blockMath-geZHyGHxm",{"__TSPROSE_proseElement":174,"schema":8717,"children":8718,"id":8733},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[8719,8721,8725,8727,8731],{"__TSPROSE_proseElement":174,"schema":8720,"data":8473},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":8722,"data":8723,"storageKey":1077,"id":8724},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":1076},"inlinerMath-8Y3Z60Vbg-15",{"__TSPROSE_proseElement":174,"schema":8726,"data":522},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":8728,"data":8729,"storageKey":806,"id":8730},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":805},"inlinerMath-fBsW71uIl-14",{"__TSPROSE_proseElement":174,"schema":8732,"data":531},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"paragraph-dHSYpJeSD-1",{"__TSPROSE_proseElement":174,"schema":8735,"children":8736},{"name":2612,"type":240,"linkable":241,"__TSPROSE_schema":174},[8737,8757,8763],{"__TSPROSE_proseElement":174,"schema":8738,"children":8739,"id":8756},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[8740,8743,8747,8749,8753],{"__TSPROSE_proseElement":174,"schema":8741,"data":8742},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"Большие трудности может создать специально подобранная перемудрённая формулировка задачи. Может показаться, что стороны по отдельности пропорциональны числам ",{"__TSPROSE_proseElement":174,"schema":8744,"data":8745,"storageKey":790,"id":8746},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":789},"inlinerMath-MAD3UaxRg-11",{"__TSPROSE_proseElement":174,"schema":8748,"data":522},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},{"__TSPROSE_proseElement":174,"schema":8750,"data":8751,"storageKey":605,"id":8752},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":604},"inlinerMath-ZXyaHjumL-13",{"__TSPROSE_proseElement":174,"schema":8754,"data":8755},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},", но с разными коэффициентами.","paragraph-pNwIJZu4r",{"__TSPROSE_proseElement":174,"schema":8758,"data":8759,"storageKey":8761,"id":8762},{"name":453,"type":240,"linkable":174,"__TSPROSE_schema":174},{"katex":8760,"freeze":241},"a = 5x >>{big} b = 4y >>{big} x \\neq y","$$ a = 5x >>{big} b = 4y >>{big} x \\neq y $$","blockMath-FRGonTwi1",{"__TSPROSE_proseElement":174,"schema":8764,"children":8765,"id":8777},{"name":245,"type":240,"linkable":174,"__TSPROSE_schema":174},[8766,8769,8775],{"__TSPROSE_proseElement":174,"schema":8767,"data":8768},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"На самом деле понимать это надо буквально -- что можно составить пропорцию ",{"__TSPROSE_proseElement":174,"schema":8770,"data":8771,"storageKey":8773,"id":8774},{"name":351,"type":250,"linkable":174,"__TSPROSE_schema":174},{"katex":8772},"a : b = 5 : 4","$ a : b = 5 : 4 $","inlinerMath-sUhPmojzI",{"__TSPROSE_proseElement":174,"schema":8776,"data":531},{"name":249,"type":250,"linkable":241,"__TSPROSE_schema":174},"paragraph-w6BNckrVW","zagadochnye-kvadraty",{"$ 40 $":8780,"$ 30 $":8780,"$ 936 $":8780,"content\u002F01-foundations\u002F02-equations\u002F03-quadratic\u002F6-real-world\u002Fassets\u002Fbox-sketch.svg":8781,"$ 2 $":8780,"$ x $":8780,"$$ 40 - x - x = 40 - 2x $$":8780,"$ 30 - 2x $":8780,"$$ (40 - 2x)(30 - 2x) = 936 \\\\ (20 - x)(30 - 2x) = 468 \\\\ (20 - x)(15 - x) = 234 \\\\ 300 - 35x + x^2 = 234 \\\\ x^2 - 35x + 66 = 0 $$":8780,"\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Ffactoring\u002Farticle\u002F$manualFactoring":8785,"$ -35 $":8780,"$ 66 $":8780,"$ -33 $":8780,"$ -2 $":8780,"$$ (x - 33)(x - 2) = 0 $$":8780,"$ 33 $":8780,"$ 120 $":8780,"$ 4 $":8780,"$ 6 $":8780,"content\u002F01-foundations\u002F02-equations\u002F03-quadratic\u002F6-real-world\u002Fassets\u002Fsteel-frame.svg":8791,"$ 3 $":8780,"$ 4 + 2x $":8780,"$ 6 + 2x $":8780,"$$ S = (4 + 2x)(6 + 2x) $$":8780,"$$ 120 = (4 + 2x)(6 + 2x) $$":8780,"$$ 120 = 24 + 8x + 12x + 4x^2 \\\\ 120 = 24 + 20x + 4x^2 $$":8780,"$$ 120 - 120 = 24 + 20x + 4x^2 - 120 \\\\ 0 = 4x^2 + 20x - 96 $$":8780,"$$ 0 = x^2 + 5x - 24 $$":8780,"$ 5 $":8780,"$ -24 $":8780,"$ 8 $":8780,"$ -3 $":8780,"$$ (x + 8)(x - 3) = 0 $$":8780,"$ -8 $":8780,"$ 12 $":8780,"$ 2x $":8780,"$ (8 + 2x) $":8780,"$ (12 + 2x) $":8780,"$ 8 \\cdot 12 = 96 $":8780,"$ 2 \\cdot 96 = 192 $":8780,"$$ (8 + 2x)(12 + 2x) = 192 $$":8780,"$$ 96 + 16x + 24x + 4x^2 = 192 \\\\ 4x^2 + 40x + 96 = 192 \\\\ 4x^2 + 40x - 96 = 0 $$":8780,"$$ x^2 + 10x - 24 = 0 $$":8780,"$ C $":8780,"$ 10 $":8780,"$$ (x + 12)(x - 2) = 0 $$":8780,"$ x = -12 $":8780,"$ x = 2 $":8780,"$ 840 $":8780,"$ 20 $":8780,"$ N_1 $":8780,"$ N_2 = N_1 + 1 $":8780,"$ 7 $":8780,"$ N_2 $":8780,"$$ N_2 = N_1 + 1 $$":8780,"$ P $":8780,"$ P - 20 $":8780,"$$ \\frac{840}{P - 20} = \\frac{840}{P} + 1 $$":8780,"$ P(P - 20) $":8780,"$$ 840P = 840(P - 20) + P(P - 20) $$":8780,"$$ \\cancel{840P} = \\cancel{840P} - 20 \\cdot 840 + P^2 - 20P \\\\ 0 = - 20 \\cdot 840 + P^2 - 20P \\\\ 0 = P^2 - 20P - 20 \\cdot 840 $$":8780,"$ B $":8780,"\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fquadratic-formula\u002Fpractice\u002F$evenB":8795,"$$ P = -(-10) \\pm \\sqrt{(-10)^2 - (-20\\cdot 840)} = 10 \\pm \\sqrt{100 + 2 \\cdot 84 \\cdot 100} = 10 \\pm \\sqrt{100(1 + 168)} = 10 \\pm \\sqrt{100}\\sqrt{169} = 10 \\pm 130 $$":8780,"$ P = 140 $":8780,"$ P = -120 $":8780,"$$ N_1 = \\frac{840}{140} = 6 $$":8780,"$ N_2 = 7 $":8780,"$ 56 $":8780,"$ 11 $":8780,"$ p $":8780,"$ a $":8780,"$$ p-6 = 2(a-6)^2 $$":8780,"$$ a + 4 = \\frac{1}{4}(p + 4) \\\\ 4a + 16 = p + 4 $$":8780,"$ p - 6 $":8780,"$$ 4a + 16 - 10 = p + 4 - 10 \\\\ 4a + 6 = p - 6 $$":8780,"$ 4a + 6 $":8780,"$$ 4a + 6 = 2(a - 6)^2 \\\\ 4a + 6 = 2(a^2 - 12a + 36) \\\\ 4a + 6 = 2a^2 - 24a + 72 \\\\ 0 = 2a^2 - 28a + 66 \\\\ $$":8780,"$$ 0 = a^2 - 14a + 33 $$":8780,"$ A $":8780,"$ -11 $":8780,"$ -14 $":8780,"$$ (a - 11)(a - 3) = 0 $$":8780,"$ 4a + 6 = p - 6 $":8780,"$$ 4 \\cdot 11 + 6 = p - 6 \\\\ 44 + 6 = p - 6 \\\\ 50 = p - 6 \\\\ p = 56 $$":8780,"$ 15 $":8780,"$ 36 $":8780,"$ 1 $":8780,"$ y $":8780,"$ xy = 36 $":8780,"$$ \\frac{15}{x-1} >>{big} \\frac{10}{y-1} $$":8780,"$$ \\frac{15}{x-1} = \\frac{10}{y-1} + 1 $$":8780,"$ y = \\frac{36}{x} $":8780,"$$ \\frac{15}{x-1} = \\frac{10}{\\frac{36}{x}-1} + 1 \\\\ \\frac{15}{x-1} = \\frac{10}{\\frac{36-x}{x}} + 1 \\\\ \\frac{15}{x-1} = \\frac{10x}{36-x} + 1 $$":8780,"$ (x-1)(36-x) $":8780,"$$ 15(36-x) = 10x(x-1) + (x-1)(36-x) $$":8780,"$$ 540 - 15x = 10x^2 - 10x + 36x - x^2 - 36 + x \\\\ 540 - 15x = 9x^2 + 27x - 36 \\\\ 0 = 9x^2 + 42x - 576 $$":8780,"$$ 3x^2 + 14x - 192 = 0 $$":8780,"$ B = 14 $":8780,"$$ x = \\frac{-7 \\pm \\sqrt{7^2 - 3 \\cdot (-192)}}{3} = \\frac{-7 \\pm \\sqrt{49 + 576}}{3} = \\frac{-7 \\pm \\sqrt{625}}{3} = \\frac{-7 \\pm 25}{3} $$":8780,"$ x = 6 $":8780,"$ x = -\\frac{32}{3} $":8780,"$$ y = \\frac{36}{x} = \\frac{36}{6} = 6 $$":8780,"$$ \\frac{15}{6-1} = \\frac{15}{5} = 3 >>{big} \\frac{10}{6-1} = \\frac{10}{5} = 2 $$":8780,"$ g = 10 $":8780,"$ h_0 $":8780,"$ v_0 $":8780,"$ h $":8780,"$ t $":8780,"$$ h(t) = h_0 + v_0 t - \\frac{g}{2}t^2 $$":8780,"$ 60 $":8780,"content\u002F01-foundations\u002F02-equations\u002F03-quadratic\u002F6-real-world\u002Fassets\u002Fpisa-experiment.svg":8800,"$$ h(t) = 60 - 5t - \\frac{10}{2}t^2 = 60 - 5t - 5t^2 $$":8780,"$$ 60 - 5t - 5t^2 = 0 $$":8780,"$ -5 $":8780,"$ t^2 $":8780,"$$ t^2 + t - 12 = 0 $$":8780,"\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fquadratic-formula\u002Farticle\u002F$discriminant":8804,"$$ D = 1^2 - 4 \\cdot 1 \\cdot (-12) = 1 + 48 = 49 $$":8780,"\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fquadratic-formula\u002Farticle\u002F$formula":8807,"$$ t = \\frac{-1 \\pm \\sqrt{49}}{2 \\cdot 1} = \\frac{-1 \\pm 7}{2} $$":8780,"$ t = -4 $":8780,"$ t = 3 $":8780,"$$ h(t) = 60 + 5t - \\frac{10}{2}t^2 = 60 + 5t - 5t^2 $$":8780,"$$ 60 + 5t - 5t^2 = 0 $$":8780,"$$ t^2 - t - 12 = 0 $$":8780,"$ -1 $":8780,"$ -12 $":8780,"$ -4 $":8780,"$$ (t + 3)(t - 4) = 0 $$":8780,"$ t = -3 $":8780,"$ t = 4 $":8780,"$ 16 $":8780,"$ 1971 $":8780,"content\u002F01-foundations\u002F02-equations\u002F03-quadratic\u002F6-real-world\u002Fassets\u002Fhammer-feather.mp4":8810,"$ R_t $":8780,"$ R_1 $":8780,"$ R_2 $":8780,"$$ \\frac{1}{R_t} = \\frac{1}{R_1} + \\frac{1}{R_2} + \\dots $$":8780,"content\u002F01-foundations\u002F02-equations\u002F03-quadratic\u002F6-real-world\u002Fassets\u002Fcircuit.svg":8812,"$$ R_2 = R_1 + 3 $$":8780,"$$ \\frac{1}{2} = \\frac{1}{R_1} + \\frac{1}{R_2} $$":8780,"$$ \\frac{1}{2} = \\frac{1}{R_1} + \\frac{1}{R_1 + 3} \\\\ R_1(R_1 + 3) = 2(R_1 + 3) + 2R_1 \\\\ R_1^2 + 3R_1 = 2R_1 + 6 + 2R_1 \\\\ R_1^2 + 3R_1 = 4R_1 + 6 \\\\ R_1^2 - R_1 - 6 = 0 $$":8780,"$ -6 $":8780,"$$ (R_1 - 3)(R_1 + 2) = 0 $$":8780,"$ R_1 = 3 $":8780,"$ R_1 = -2 $":8780,"$$ R_2 = R_1 + 3 = 3 + 3 = 6 $$":8780,"$ S = V \\cdot t $":8780,"$ V = 2 $":8780,"$ t = 10 $":8780,"$ S = V \\cdot t = 2 \\cdot 10 = 20 $":8780,"$$ S = V \\cdot t >>{big} V = \\frac{S}{t} >>{big} t = \\frac{S}{V} $$":8780,"content\u002F01-foundations\u002F02-equations\u002F03-quadratic\u002F6-real-world\u002Fassets\u002Friver-ride.svg":8816,"$ V $":8780,"$ V - 2 $":8780,"$$ t_{\\rarr} = \\frac{8}{V - 2} $$":8780,"$ V + 2 $":8780,"$$ t_{\\larr} = \\frac{8}{V + 2} $$":8780,"$$ t_{\\rarr} + t_{\\larr} = 3 $$":8780,"$$ \\frac{8}{V - 2} + \\frac{8}{V + 2} = 3 \\\\ 8(V + 2) + 8(V - 2) = 3(V - 2)(V + 2) \\\\ 8V + 16 + 8V - 16 = 3(V^2 - 4) \\\\ 16V = 3V^2 - 12 \\\\ 0 = 3V^2 - 16V - 12 $$":8780,"$$ D = (-16)^2 - 4 \\cdot 3 \\cdot (-12) = 256 + 144 = 400 $$":8780,"$$ V = \\frac{16 \\pm \\sqrt{400}}{2 \\cdot 3} = \\frac{16 \\pm 20}{6} $$":8780,"$ V = 6 $":8780,"$ V = -\\frac{2}{3} $":8780,"$ -\\frac{2}{3} $":8780,"\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Freal-world\u002F$_whyNotReal":8820,"$$ t_{\\rarr} = \\frac{8}{6 - 2} = \\frac{8}{4} = 2 $$":8780,"$ 14 $":8780,"$ S = Vt $":8780,"$$ \\boxed{t_{\\text{против течения}} + t_{\\text{по течению}} = t_{\\text{стоячая}}} $$":8780,"$$ S = Vt $$":8780,"$$ \\frac{S}{V} = \\frac{\\cancel{V}t}{\\cancel{V}} >>{big} \\boxed{t = \\frac{S}{V}} $$":8780,"$$ \\frac{S_{\\text{против течения}}}{V_{\\text{против течения}}} + \\frac{S_{\\text{по течению}}}{V_{\\text{по течению}}} = \\frac{S_{\\text{стоячая}}}{V_{\\text{стоячая}}} $$":8780,"$ V_б $":8780,"$$ \\frac{15}{V_б - 1} + \\frac{14}{V_б + 1} = \\frac{30}{V_б} $$":8780,"$$ 15(V_б + 1)V_б + 14(V_б - 1)V_б = 30(V_б + 1)(V_б - 1) \\\\ 15(V_б^2 + V_б) + 14(V_б^2 - V_б) = 30(V_б^2 - 1) \\\\ 15V_б^2 + 15V_б + 14V_б^2 - 14V_б = 30V_б^2 - 30 \\\\ 29V_б^2 + V_б = 30V_б^2 - 30 \\\\ 0 = 30V_б^2 - 29V_б^2 - V_б - 30 \\\\ 0 = V_б^2 - V_б - 30 $$":8780,"$$ D = (-1)^2 - 4 \\cdot 1 \\cdot (-30) = 1 + 120 = 121 $$":8780,"$$ V_б = \\frac{-(-1) \\pm \\sqrt{121}}{2 \\cdot 1} = \\frac{1 \\pm 11}{2} $$":8780,"$ V_б = 6 $":8780,"$ V_б = -5 $":8780,"$ 200 $":8780,"$ V_1 $":8780,"$ t_1 $":8780,"$ V_2 = V_1 + 10 $":8780,"$ t_2 = t_1 - 1 $":8780,"$$ V_2 = V_1 + 10 $$":8780,"$ V = \\frac{S}{t} $":8780,"$$ \\frac{S}{t_2} = \\frac{S}{t_1} + 10 \\\\ \\frac{200}{t_1 - 1} = \\frac{200}{t_1} + 10 $$":8780,"$ t_1(t_1 - 1) $":8780,"$$ 200t_1 = 200(t_1 - 1) + 10t_1(t_1 - 1) \\\\ 200t_1 = 200t_1 - 200 + 10t_1^2 - 10t_1 \\\\ 0 = 10t_1^2 - 10t_1 - 200 \\\\ 0 = t_1^2 - t_1 - 20 $$":8780,"$ -20 $":8780,"$$ (t_1 + 4)(t_1 - 5) = 0 $$":8780,"$ t_1 = 5 $":8780,"$$ S = Vt >>{big} \\frac{S}{t} = V >>{big} V = \\frac{200}{5} = 40 $$":8780,"$ \\frac{S}{V} $":8780,"$ 1.5 $":8780,"$ t_{\\uarr} $":8780,"$ t_{\\darr} $":8780,"$ V_\\approx $":8780,"$$ t_{\\darr} = t_{\\uarr} - \\underbrace{1.5}_{\\small\\frac{3}{2}} $$":8780,"$$ \\frac{36}{10 + V_\\approx} = \\frac{36}{10 - V_\\approx} - \\frac{3}{2} \\\\ 36(10 - V_\\approx) = 36(10 + V_\\approx) - \\frac{3}{2}(10 + V_\\approx)(10 - V_\\approx) \\\\ \\cancel{360} - 36V_\\approx = \\cancel{360} + 36V_\\approx - \\frac{3}{2}(100 - V_\\approx^2) \\\\ -72V_\\approx = - \\frac{3}{2}(100 - V_\\approx^2) \\\\ -24V_\\approx = -\\frac{1}{2}(100 - V_\\approx^2) \\\\ 48V_\\approx = 100 - V_\\approx^2 \\\\ V_\\approx^2 + 48V_\\approx - 100 = 0 $$":8780,"$$ (V_\\approx + 50)(V_\\approx - 2) = 0 $$":8780,"$ -50 $":8780,"$ t_2 = t_1 - 2 $":8780,"$$ V_2 = V_1 + 1 $$":8780,"$$ \\frac{40}{t_2} = \\frac{40}{t_1} + 1 $$":8780,"$ t_2 $":8780,"$$ \\frac{40}{t_1 - 2} = \\frac{40}{t_1} + 1 $$":8780,"$ t_1(t_1 - 2) $":8780,"$$ 40t_1 = 40(t_1 - 2) + t_1(t_1 - 2) \\\\ 40t_1 = 40t_1 - 80 + t_1^2 - 2t_1 \\\\ 0 = t_1^2 - 2t_1 - 80 $$":8780,"$ -80 $":8780,"$ -10 $":8780,"$$ (t_1 + 8)(t_1 - 10) = 0 $$":8780,"$ t_1 = -8 $":8780,"$ t_1 = 10 $":8780,"$ 18 $":8780,"$ 24 $":8780,"$ V + 3 $":8780,"$ 2V $":8780,"$ 2(V + 3) = 2V + 6 $":8780,"$ 2V + 6 $":8780,"$$ (2V)^2 + (2V + 6)^2 = 30^2 $$":8780,"$$ 4V^2 + 4V^2 + 24V + 36 = 900 \\\\ 8V^2 + 24V + 36 = 900 \\\\ 8V^2 + 24V - 864 = 0 $$":8780,"$$ V^2 + 3V - 108 = 0 $$":8780,"$ -108 $":8780,"$ -9 $":8780,"$$ (V + 12)(V - 9) = 0 $$":8780,"$ V = -12 $":8780,"$ V = 9 $":8780,"$ 9 $":8780,"$$ d_{\\text{восток}} = 2 \\cdot 9 = 18 >>{big} d_{\\text{север}} = 2 \\cdot 12 = 24 $$":8780,"$ S $":8780,"$ P = 8 $":8780,"$ 8 \\cdot 10 = 80 = A $":8780,"$ A = Pt $":8780,"$$ A = Pt >>{big} P = \\frac{A}{t} >>{big} t = \\frac{A}{P} $$":8780,"$$ A = (P_1 + P_2) \\cdot 6 $$":8780,"$ P = \\frac{A}{t} $":8780,"$$ A = \\left(\\frac{A}{t_1} + \\frac{A}{t_2}\\right) \\cdot 6 $$":8780,"$$ \\cancel{A} = \\cancel{A} \\cdot \\left(\\frac{1}{t_1} + \\frac{1}{t_2}\\right) \\cdot 6 \\\\ 1 = \\left(\\frac{1}{t_1} + \\frac{1}{t_2}\\right) \\cdot 6 \\\\ \\frac{1}{6} = \\frac{1}{t_1} + \\frac{1}{t_2} $$":8780,"$ t_1 = t_2 - 5 $":8780,"$$ \\frac{1}{6} = \\frac{1}{t_2 - 5} + \\frac{1}{t_2} $$":8780,"$$ \\frac{(t_2 - 5)t_2}{6} = t_2 + t_2 - 5 \\\\ \\frac{t_2^2 - 5t_2}{6} = 2t_2 - 5 \\\\ t_2^2 - 5t_2 = 12t_2 - 30 \\\\ t_2^2 - 17t_2 + 30 = 0 $$":8780,"$ A=1 $":8780,"$ -15 $":8780,"$ -17 $":8780,"$$ (t_2 - 15)(t_2 - 2) = 0 $$":8780,"$ t_1 = t_2 - 5 = -3 $":8780,"$ 240 $":8780,"$ 35% $":8780,"$ 27 $":8780,"$ A_1 = 240 $":8780,"$$ A_2 = A_1 + \\frac{35}{100} \\cdot A_1 = 240 + \\frac{35}{100} \\cdot 240 = 324 $$":8780,"$$ P_1 = P_2 - 3 >>{big} t_1 = t_2 - 2 $$":8780,"$$ \\frac{A_1}{P_1} = \\frac{A_2}{P_2} - 2 $$":8780,"$ P_1 $":8780,"$ P_2 $":8780,"$$ \\frac{240}{P_2 - 3} = \\frac{324}{P_2} - 2 \\\\ 240P_2 = 324(P_2 - 3) - 2P_2(P_2 - 3) \\\\ 120P_2 = 162(P_2 - 3) - P_2(P_2 - 3) \\\\ 120P_2 = 162P_2 - 486 - P_2^2 + 3P_2 \\\\ P_2^2 - 45P_2 + 486 = 0 $$":8780,"$$ D = (-45)^2 - 4 \\cdot 1 \\cdot 486 = 2025 - 1944 = 81 $$":8780,"$$ P_2 = \\frac{-(-45) \\pm \\sqrt{81}}{2 \\cdot 1} = \\frac{45 \\pm 9}{2} $$":8780,"$ P_2 = 27 $":8780,"$ P_2 = 18 $":8780,"$ P_1 = 27 - 3 = 24 $":8780,"$ 3\u002F4 $":8780,"$ \\frac{56}{3} $":8780,"$$ A = (P_1 + P_2 + P_3) \\cdot 6 $$":8780,"$$ t_2 = \\frac{3}{4}t_1 >>{big} t_3 = t_2 + 10 $$":8780,"$$ A = \\left(\\frac{A}{t_1} + \\frac{A}{t_2} + \\frac{A}{t_3}\\right) \\cdot 6 $$":8780,"$$ 1 = \\left(\\frac{1}{t_1} + \\frac{1}{t_2} + \\frac{1}{t_3}\\right) \\cdot 6 $$":8780,"$$ t_2 = \\frac{3}{4}t_1 \\implies t_1 = \\frac{4}{3}t_2 $$":8780,"$ t_3 $":8780,"$$ 1 = \\left(\\frac{1}{\\frac{4}{3}t_2} + \\frac{1}{t_2} + \\frac{1}{t_2 + 10}\\right) \\cdot 6 \\\\ \\frac{1}{6} = \\frac{3}{4t_2} + \\frac{1}{t_2} + \\frac{1}{t_2 + 10} \\\\ \\frac{1}{6} = \\frac{7}{4t_2} + \\frac{1}{t_2 + 10} $$":8780,"$ 12t_2(t_2 + 10) $":8780,"$$ 2t_2(t_2 + 10) = 21(t_2 + 10) + 12t_2 \\\\ 2t_2^2 + 20t_2 = 21t_2 + 210 + 12t_2 \\\\ 2t_2^2 - 13t_2 - 210 = 0 $$":8780,"$$ D = (-13)^2 - 4 \\cdot 2 \\cdot (-210) = 169 + 1680 = 1849 $$":8780,"$$ t_2 = \\frac{-(-13) \\pm \\sqrt{1849}}{2 \\cdot 2} = \\frac{13 \\pm 43}{4} $$":8780,"$ t_2 = 14 $":8780,"$ t_2 = -7.5 $":8780,"$$ t_1 = \\frac{4}{3} \\cdot 14 = \\frac{56}{3} \\text{ мин} >>{big} t_3 = 14 + 10 = 24 \\text{ мин} $$":8780,"$ c $":8780,"$$ c_A = \\frac{m_A}{M} >>{big} c_A = \\frac{V_A}{V} $$":8780,"$ 3% $":8780,"$ c = 0.03 $":8780,"$ 600 $":8780,"$ 4% $":8780,"$ 0.06 $":8780,"$ 6% $":8780,"$ M $":8780,"$ с $":8780,"$$ c = \\frac{18}{M} $$":8780,"$ 0.04 $":8780,"$$ c - 0.04 = \\frac{18}{M + 600} $$":8780,"$$ \\frac{18}{M} - 0.04 = \\frac{18}{M + 600} \\\\ \\frac{18}{M} - \\frac{4}{100} = \\frac{18}{M + 600} \\\\ \\frac{9}{M} - \\frac{2}{100} = \\frac{9}{M + 600} \\\\ 900(M+600) - 2M(M+600) = 900M \\\\ 450(M+600) - M(M+600) = 450M \\\\ \\cancel{450M} + 450 \\cdot 600 - M^2 - 600M = \\cancel{450M} \\\\ 450 \\cdot 600 - M^2 - 600M = 0 \\\\ M^2 + 600M - 450 \\cdot 600 = 0 $$":8780,"$$ M = -300 \\pm \\sqrt{300^2 + 450 \\cdot 600} = -300 \\pm 600 $$":8780,"$ M = 300 $":8780,"$ M = -900 $":8780,"$$ c = \\frac{18}{300} = \\frac{6}{100} = 0.06 $$":8780,"$ 40% $":8780,"$ 36% $":8780,"$ m_1 + m_2 = M $":8780,"$ m_1 $":8780,"$ m_2 $":8780,"$ 20% $":8780,"$$ 0.36 = \\frac{6 + 12}{M} \\implies M = \\frac{18}{0.36} = \\frac{18 \\cdot 100}{36} = 50 $$":8780,"$$ c_1 = c_2 - 0.4 $$":8780,"$$ m_1 + m_2 = 50 $$":8780,"$$ c_1 = \\frac{6}{m_1} \\implies m_1 = \\frac{6}{c_1} >>{big} c_2 = \\frac{12}{m_2} \\implies m_2 = \\frac{12}{c_2} $$":8780,"$$ \\frac{6}{c_1} + \\frac{12}{c_2} = 50 $$":8780,"$$ \\frac{6}{c_2 - 0.4} + \\frac{12}{c_2} = 50 $$":8780,"$$ \\frac{6}{c_2 - \\frac{4}{10}} + \\frac{12}{c_2} = 50 \\\\ \\frac{3}{c_2 - \\frac{4}{10}} + \\frac{6}{c_2} = 25 \\\\ 3c_2 + 6\\left(c_2 - \\frac{4}{10}\\right) = 25c_2\\left(c_2 - \\frac{4}{10}\\right) \\\\ 3c_2 + 6c_2 - \\frac{24}{10} = 25c_2^2 - 10c_2 \\\\ 0 = 25c_2^2 - 19c_2 + \\frac{12}{5} $$":8780,"$$ D = (-19)^2 - 4 \\cdot 25 \\cdot \\frac{12}{5} = 361 - 240 = 121 $$":8780,"$$ c_2 = \\frac{-(-19) \\pm \\sqrt{121}}{2 \\cdot 25} = \\frac{19 \\pm 11}{50} $$":8780,"$ c_2 = 0.6 $":8780,"$ c_2 = 0.16 $":8780,"$ c_1 = -0.24 $":8780,"$ c_1 = 0.2 $":8780,"$ 16% $":8780,"$ 9% $":8780,"$ 8-x $":8780,"$ (8-x) \\cdot 0.16 $":8780,"$$ \\frac{(8-x) \\cdot 0.16}{8} = (8-x) \\cdot 0.02 $$":8780,"$ 0.16 - 0.02x $":8780,"$ (8-x) \\cdot (8-x) \\cdot 0.02 $":8780,"$ 0.09 $":8780,"$$ \\frac{(8-x) \\cdot (8-x) \\cdot 0.02}{8} = 0.09 \\\\ \\frac{(8-x)^2 \\cdot 0.02}{8} = 0.09 \\\\ $$":8780,"$ 100 $":8780,"$$ \\frac{(8-x)^2 \\cdot 2}{8} = 9 \\\\ (8-x)^2 = 36 \\\\ 8-x = \\pm 6 \\\\ 8\\mp 6 = x $$":8780,"$ x = 14 $":8780,"$ n $":8780,"$ 1, 2, 3\\ldots $":8780,"$$ S = \\frac{1}{2}n(n+1) $$":8780,"$ 465 $":8780,"$$ 465 = \\frac{1}{2}n(n+1) \\\\ 930 = n(n+1) \\\\ 930 = n^2 + n \\\\ 0 = n^2 + n - 930 $$":8780,"$$ D = 1^2 - 4 \\cdot 1 \\cdot (-930) = 1 + 3720 = 3721 $$":8780,"$$ n = \\frac{-1 \\pm \\sqrt{3721}}{2 \\cdot 1} = \\frac{-1 \\pm 61}{2} $$":8780,"$ n = 30 $":8780,"$ n = -31 $":8780,"$ -31 $":8780,"$ 22 $":8780,"$ m $":8780,"$$ n + m = 22 >>{big} n\\cdot m = 120 $$":8780,"$$ n = 22 - m $$":8780,"$$ (22 - m)m = 120 \\\\ 22m - m^2 = 120 \\\\ 0 = m^2 - 22m + 120 $$":8780,"$ A = 1 $":8780,"$ -22 $":8780,"$$ (m - 10)(m - 12) = 0 $$":8780,"$ m = 10 $":8780,"$ m = 12 $":8780,"$ n = 22 - m $":8780,"$ n = 12 $":8780,"$ n = 10 $":8780,"$$ 12 = 1\\cdot 10 + 2 >>{big} 84 = 8\\cdot 10 + 4 $$":8780,"$ 26 $":8780,"$ b $":8780,"$ 1\\cdot 10 + 2 $":8780,"$ 84 $":8780,"$ 8\\cdot 10 + 4 $":8780,"$$ ab = 12 $$":8780,"$$ 10a + b + 36 = 10b + a \\\\ 9a + 36 = 9b \\\\ a + 4 = b $$":8780,"$ ab = 12 $":8780,"$$ a(a + 4) = 12 \\\\ a^2 + 4a - 12 = 0 $$":8780,"$$ (a + 6)(a - 2) = 0 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\\\\ a^2 = 9 \\\\ a = \\pm 3 $$":8780,"$ 50 $":8780,"$$ a \\cdot b = 50 >>{big}{bottom} a + b + a + b = 30 \\\\ 2(a+b) = 30 $$":8780,"$ a = 50\u002Fb $":8780,"$$ 2\\left(\\frac{50}{b} + b\\right) = 30 \\\\ \\frac{100}{b} + 2b = 30 \\\\ 100 + 2b^2 = 30b \\\\ 2b^2 - 30b + 100 = 0 \\\\ b^2 - 15b + 50 = 0 $$":8780,"$$ (b - 10)(b - 5) = 0 $$":8780,"$ b = 10 $":8780,"$ b = 5 $":8780,"$$ a = \\frac{50}{10} = 5 >>{big} a = \\frac{50}{5} = 10 $$":8780,"$ 28 $":8780,"$$ \\frac{a}{b} = \\frac{5}{4} $$":8780,"$ (a - 2)^2 $":8780,"$ (b - 2)^2 $":8780,"$$ a = \\frac{5}{4}b $$":8780,"$$ (a - 2)^2 - (b - 2)^2 = 28 $$":8780,"$$ (a - 2 - (b - 2))(a - 2 + (b - 2)) = 28 \\\\ (a - b)(a + b - 4) = 28 $$":8780,"$ \\frac{5}{4}b $":8780,"$$ \\left(\\frac{5}{4}b - b\\right)\\left(\\frac{5}{4}b + b - 4\\right) = 28 \\\\ \\left(\\frac{1}{4}b\\right)\\left(\\frac{9}{4}b - 4\\right) = 28 \\\\ \\frac{9}{16}b^2 - b = 28 \\\\ 9b^2 - 16b = 448 \\\\ 9b^2 - 16b - 448 = 0 $$":8780,"$$ b = \\frac{-(-8) \\pm \\sqrt{(-8)^2 - 9 \\cdot (-448)}}{9} = \\frac{8 \\pm \\sqrt{64 + 4032}}{9} = \\frac{8 \\pm \\sqrt{4096}}{9} = \\frac{8 \\pm 64}{9} $$":8780,"$ b = -\\frac{56}{9} $":8780,"$$ a = \\frac{5}{4}b = \\frac{5}{4} \\cdot 8 = 10 $$":8780,"$$ a = 5x >>{big} b = 4y >>{big} x \\neq y $$":8780,"$ a : b = 5 : 4 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