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Консультация № 174783

02.12.2009, 21:06

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Образование

Помогите решить задачу:
Боковые рёбра правильной усечённой треугольной пирамиды наклонены к плоскости основания под углом α. Сторона нижнего основания равна a, а верхнего . Найти объём усечённой пирамиды.

Заранее спасибо.

Обсуждение

Неизвестный

02.12.2009, 21:09

общий

Сторона нижнего основания равна A, а верхнего B.(A>B)

давно

Гордиенко Андрей Владимирович

Мастер-Эксперт
17387
18345

05.12.2009, 09:53

общий

это ответ

Здравствуйте, Arkalis.

Для объема усеченной пирамиды известна формула
V = 1/3 ∙ H ∙ (S₁ + S₂ + √(S₁S₂)), (1)
где H – высота пирамиды; S₁, S₂ – площади оснований.

Выполним рисунок, на котором изобразим пирамиду и вид не нее сверху.

Из рисунка видно, что tg α = H/(R – r),
H = (R – r)tg α = (a/√3 – b/√3)tg α = 1/√3 ∙ (a – b)tg α. (2)

Имеем также
S₁ = a³/(4R) = a³/(4a/√3) = a²√3/4, (3)
S₂ = b³/(4r) = b³/(4b/√3) = b²√3/4. (4)

Подставляя выражения (2) – (4) в формулу (1), получаем
V = 1/3 ∙ 1/√3 ∙ (a – b)tg α ∙ (a²√3/4 + b²√3/4 + √(a²√3/4 ∙ b²√3/4)) =
= 1/12 ∙ (a – b)tg α ∙ (a² + b² + ab).

Ответ: V = 1/12 ∙ (a – b)tg α ∙ (a² + b² + ab).

С уважением.

Об авторе:
Facta loquuntur.

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