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Консультация № 177673
04.04.2010, 19:46
0.00 руб.
0
1
1
здравствуйте,уважаемые эксперты.помогите пожалуйста решить задачу:
высота конуса разделена на три равных отрезка и через точки деления параллельно основанию проведены плоскости,разбивающие конус на три части.найти объём среднего усеченного конуса,если объём данного конуса равен V
заранее благодарен.
высота конуса разделена на три равных отрезка и через точки деления параллельно основанию проведены плоскости,разбивающие конус на три части.найти объём среднего усеченного конуса,если объём данного конуса равен V
заранее благодарен.
Обсуждение
Неизвестный
04.04.2010, 20:45
общий
это ответ
Здравствуйте, G-buck.
Рассмотрим диаметральное сечение конуса
R - радиус основания исходного конуса
H - высота исходного конуса
Обозначим:
[$945$] - угол наклона образующей
r1 - радиус конуса высотой 2*H/3
r2 - радиус конуса высотой H/3
Vук - объем усеченного конуса (среднего)
Vk=(1/3)*Pi*R2*H = V
sin([$945$])= H/R = (H/3+H/3)/r1 = (H/3)/r2
Получим
r1= (H/3+H/3)/(H/R)=(2/3)*R
r2= (1/3)*R
искомый объем Vук=(1/3)*Pi*(r12+r1*r2+r22)*(H/3)=
=(1/3)*Pi*R2*H*(7/27)
Получим Vук = (7/27)*V
Ответ: (7/27)*V
Рассмотрим диаметральное сечение конуса
R - радиус основания исходного конуса
H - высота исходного конуса
Обозначим:
[$945$] - угол наклона образующей
r1 - радиус конуса высотой 2*H/3
r2 - радиус конуса высотой H/3
Vук - объем усеченного конуса (среднего)
Vk=(1/3)*Pi*R2*H = V
sin([$945$])= H/R = (H/3+H/3)/r1 = (H/3)/r2
Получим
r1= (H/3+H/3)/(H/R)=(2/3)*R
r2= (1/3)*R
искомый объем Vук=(1/3)*Pi*(r12+r1*r2+r22)*(H/3)=
=(1/3)*Pi*R2*H*(7/27)
Получим Vук = (7/27)*V
Ответ: (7/27)*V
5
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