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Консультация № 195868
19.06.2019, 16:48
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Здравствуйте! У меня возникли сложности с таким вопросом:
найдите апофему правильной четырехугольной пирамиды которая имеет объем 24 дм^3 и высота 8 за
найдите апофему правильной четырехугольной пирамиды которая имеет объем 24 дм^3 и высота 8 за
Обсуждение
19.06.2019, 16:56
общий
Адресаты:
sohratgylyjow24 и высота 8 за
Уточните, пожалуйста, что Вы имеете в виду.
Об авторе:
Facta loquuntur.
Facta loquuntur.
20.06.2019, 14:43
общий
это ответ
Здравствуйте, sohratgylyjow24!
Пусть правильная четырёхугольная пирамида имеет объём V=24 дм3 и высоту H=8 дм. Согласно [1, с. 13], для объёма пирамиды имеет место формула V=SH/3, где S -- площадь основания. Обозначим через [$945$] двугранный угол при основании пирамиды, через l -- искомую апофему. Тогда S=a2, где a -- сторона основания пирамиды (квадрата). С другой стороны, S=3V/H. поэтому a2=3V/H, a=[$8730$](3V/H). Но tg[$945$]=2H/a, [$945$]=arctg(2H/a),
(мы использовали формулу cos(arctg(x))=1/[$8730$](1+x2) [2, с. 328]).
Литература
1. Цыпкин А. Г., Цыпкин Г. Г. Математические формулы. -- М.: Наука, 1985. -- 128 с.
2. Зайцев В. В. и др. Элементарная математика. -- М.: Наука, 1976. -- 592 с.
Пусть правильная четырёхугольная пирамида имеет объём V=24 дм3 и высоту H=8 дм. Согласно [1, с. 13], для объёма пирамиды имеет место формула V=SH/3, где S -- площадь основания. Обозначим через [$945$] двугранный угол при основании пирамиды, через l -- искомую апофему. Тогда S=a2, где a -- сторона основания пирамиды (квадрата). С другой стороны, S=3V/H. поэтому a2=3V/H, a=[$8730$](3V/H). Но tg[$945$]=2H/a, [$945$]=arctg(2H/a),
l=a/(2*cos[$945$])=a/(2*cos(arctg(2H/a)))=([$8730$](3V/H))/(2*cos(arctg(2H/[$8730$](3V/H))))=
=([$8730$](3*24/8))/(2*cos(arctg(2*8/[$8730$](3*24/8))))=3/2*[$8730$](1+(16/3)2)=[$8730$]265/2[$8776$]8,14 (дм)
(мы использовали формулу cos(arctg(x))=1/[$8730$](1+x2) [2, с. 328]).
Литература
1. Цыпкин А. Г., Цыпкин Г. Г. Математические формулы. -- М.: Наука, 1985. -- 128 с.
2. Зайцев В. В. и др. Элементарная математика. -- М.: Наука, 1976. -- 592 с.
Об авторе:
Facta loquuntur.
Facta loquuntur.
дм^3 ,
дм 
--?
Значит 
По теореме Пифагора 
дм
Форма ответа
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