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Задать вопрос экспертам

Консультация № 182049

30.01.2011, 15:56

50.05 руб.

0 1 1

Образование

Здравствуйте! У меня возникли сложности с таким вопросом:
Сосуд представляет собой круглый прямой конус, замкнутый круглой крышкой с диаметром, равным диаметру основания конуса.
Полная поверхность сосуда равна S. Найти наибольший объем сосуда.

Обсуждение

Неизвестный

30.01.2011, 16:24

общий

это ответ

Здравствуйте, Марина!
Площадь поверхности сосуда
S=S_конуса+S_крышки=[$960$][$183$]R[$183$]L+[$960$][$183$]R²
Объем сосуда
V=1/3 [$183$][$960$]R²[$183$]H
H - высота конуса
L - образующая
H=[$8730$](L²-R²)
Из выражения для площади поверхности сосуда выражаем длину образующей через радиус
L=(S-[$960$]R²)/([$960$]R)
Тогда H=[$8730$](S²-2[$960$]R²S) / ([$960$]R)
V(R)=1/3 R [$8730$](S²-2[$960$]R²S)
Находим производную
V'(R)=1/3 [$8730$](S²-2[$960$]R²S) - 2/3 [$960$]R²S / [$8730$](S²-2[$960$]R²S)
V'(R)=0
1/3 [$8730$](S²-2[$960$]R²S) - 2/3 [$960$]R²S / [$8730$](S²-2[$960$]R²S)=0
S²-2[$960$]R²S >0
R<[$8730$](S/2[$960$])
S²-2[$960$]R²S - 2[$960$]R²S =0
S²-4[$960$]R²S =0
R=[$8730$](S/4[$960$])
При R<[$8730$](S/4[$960$]) V'(R)>0
При R>[$8730$](S/4[$960$]) V'(R)<0
=> R=[$8730$](S/4[$960$]) - точка локального максимума
V([$8730$](S/4[$960$]))=1/12 S[$8730$](2S/[$960$])

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