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Консультация № 144293
18.09.2008, 13:36
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Эксперты, помогите,пожалуйста, решить задачку.
Из пункта A вверх по реке отправилась моторная лодка, а из пункта B, находящегося выше пункта A, одновременно вышел плот. Через a часов они встретились и далее двигались без остановок. Дойдя до B, лодка, не задерживаясь, повернула обратно и догнала плот в пункте A. Сколько времени плыли плот и лодка до встречи в пункте A?
Из пункта A вверх по реке отправилась моторная лодка, а из пункта B, находящегося выше пункта A, одновременно вышел плот. Через a часов они встретились и далее двигались без остановок. Дойдя до B, лодка, не задерживаясь, повернула обратно и догнала плот в пункте A. Сколько времени плыли плот и лодка до встречи в пункте A?
Обсуждение
18.09.2008, 15:34
общий
это ответ
Здравствуйте, Daiger!
Пусть S - расстояние от A до B, v - скорость лодки в стоячей воде, v0 - скорость течения реки. До пункта A плот плыл T = S/v0 часов. До первой встречи с плотом моторная лодка прошла путь (v-v0)*a, плот прошёл путь v0*a, в сумме эти две величины должны дать расстояние от A до B: S = (v-v0)*a + v0*a, откуда S = v*a.
Таким образом
T = a*(v/v0) = a*x, где x = v/v0.
Моторная лодка плыла до пункта B и обратно столько же времени, сколько плыл плот от B до A:
S/(v-v0) + S/(v+v0) = S/v0,
то есть:
1/(v0*x-v0) + 1/(v0*x+v0) = 1/v0, или 1/(x-1)+1/(x+1) = 1,
2x = x^2 - 1.
Положительный корень этого квадратного уравнения равен:
x = 1 + sqrt(2),
следовательно
T = a*(1+sqrt(2)).
Пусть S - расстояние от A до B, v - скорость лодки в стоячей воде, v0 - скорость течения реки. До пункта A плот плыл T = S/v0 часов. До первой встречи с плотом моторная лодка прошла путь (v-v0)*a, плот прошёл путь v0*a, в сумме эти две величины должны дать расстояние от A до B: S = (v-v0)*a + v0*a, откуда S = v*a.
Таким образом
T = a*(v/v0) = a*x, где x = v/v0.
Моторная лодка плыла до пункта B и обратно столько же времени, сколько плыл плот от B до A:
S/(v-v0) + S/(v+v0) = S/v0,
то есть:
1/(v0*x-v0) + 1/(v0*x+v0) = 1/v0, или 1/(x-1)+1/(x+1) = 1,
2x = x^2 - 1.
Положительный корень этого квадратного уравнения равен:
x = 1 + sqrt(2),
следовательно
T = a*(1+sqrt(2)).
Форма ответа
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