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Задать вопрос экспертам

Консультация № 177589

01.04.2010, 03:31

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Образование

здравствуйте,уважаемые эксперты.помогите пожалуйста решить задачу:
определить угол между высотой и образующей конуса,боковая поверхность которого делится на две равновеликие части линией пересечения её со сферической поверхностью,имеющий центр в вершине конуса и радиусом высоту конуса

заранее благодарен

Обсуждение

Неизвестный

01.04.2010, 13:42

общий

это ответ

Здравствуйте, G-buck.

Линией пересечения конуса и сферы будет окружность, поэтому боковая поверхность конуса будет состоять из боковой поверхности конуса, лежащего внутри сферы и боковой поверхности усеченного конуса.

[$945$] - искомый угол
R - радиус сферы
H - высота конуса
R=H
L- образующая конуса
sin([$945$])=r₁/R = r/L получим
r₁=(R*r)/L
r=[$8730$](L²-R²)
r₁=(R/L)*[$8730$](L²-R²)

Площадь боковой поверхности внутреннего конуса - Pi*r₁*R = Pi*(R²/L)*[$8730$](L²-R²)
Площадь боковой поверхности усеченного конуса - Pi*(r+r₁)*(L-R) = Pi*([$8730$](L²-R²)+(R/L)*[$8730$](L²-R²))*(L-R)
по условию эти площади равны
Pi*(R²/L)*[$8730$](L²-R²)= Pi*([$8730$](L²-R²)+(R/L)*[$8730$](L²-R²))*(L-R)
Сокращая и раскрывая скобки получим
R²=L²-R²
L²=2*R²
L=[$8730$]2*R
sin([$945$])=r/L=[$8730$](L²-R²)/L=R/([$8730$]2*R)=1/[$8730$]2=[$8730$]2/2
[$945$]=Pi/4 (45^o)
Ответ: Pi/4

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