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Консультация № 188843

22.02.2016, 20:01

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Образование

Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос: В шар, площадь поверхности которого равна S, вписан конус. Вычислить объём конуса, если угол при вершине его осевого сечения равен 2a. Заранее благодарен.

Обсуждение

давно

Лысков Игорь Витальевич

Посетитель
7438
7205

23.02.2016, 02:47

общий

это ответ

Здравствуйте, Тимофеев Алексей Валентинович!
Пусть радиус шара R. Тогда площадь поверхности S=4[font=Times][$960$][/font]R², откуда R=(1/2)[$8730$](S/[font=Times][$960$][/font])
Пусть радиус основания конуса r, а высота h. Тогда объем конуса V=(1/3)[font=Times][$960$][/font]r²h

Применительно к рисунку, имеем:
|OB|=|OC|=R, |DC|=|AD|=r, |BO|+|OD|=h, [$8736$]ABC=2a.
Треугольник ABC - равнобедренный, BD - высота [$8658$] [$8736$]ABD=[$8736$]DBC=2a/2=a
Треугольник BOC также равнобедренный [$8658$] [$8736$]BCO=a
Очевидно, что [$8736$]BCD=90[$176$]-a и [$8736$]OCD=90[$176$]-2a
Из [$8895$]ODC:
r=|DC|=|OC|cos([$8736$]OCD)=Rcos(90[$176$]-2a)=Rsin(2a)
|OD|=|OC|sin([$8736$]OCD)=Rsin(90[$176$]-2a)=Rcos(2a) [$8658$] h=R+Rcos(2a)=R(1+cos(2a))
Тогда V=(1/3)[font=Times][$960$][/font]r²h=(1/3)[font=Times][$960$][/font]R²sin²(2a)R(1+cos(2a))=(1/3)[font=Times][$960$][/font]R³sin²(2a)(1+cos(2a))
V=(1/3)[font=Times][$960$][/font]((1/2)[$8730$](S/[font=Times][$960$][/font]))³sin²(2a)(1+cos(2a))=(S/24)[$8730$](S/[font=Times][$960$][/font])sin²(2a)(1+cos(2a))

5

Об авторе:
"Если вы заметили, что вы на стороне большинства, —
это верный признак того, что пора меняться." Марк Твен

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