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Консультация № 197694
04.02.2020, 23:06
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Уважаемые эксперты! Пожалуйста, ответьте на вопрос:
На рисунке 25, а, бизображена правильная четырёхугольная призма АВСDA1B1C1D1. Точка О - точка пересечения диагоналей грани АВСD. Угол наклона отрезка А1О к плоскости DBB1 равен 60 градусов. Вычислите объём призмы, если известно, что длина диагонали основания равна 2 корня из 3 см.
На рисунке 25, а, бизображена правильная четырёхугольная призма АВСDA1B1C1D1. Точка О - точка пересечения диагоналей грани АВСD. Угол наклона отрезка А1О к плоскости DBB1 равен 60 градусов. Вычислите объём призмы, если известно, что длина диагонали основания равна 2 корня из 3 см.
Обсуждение
14.02.2020, 04:03
общий
это ответ
Здравствуйте, svrvsvrv!
Объём призмы равен произведению площади основания на высоту. В данном случае основанием правильной четырёхугольной призмы является квадрат с диагональю 2[$8730$]3 см, поэтому площадь основания будет равна (2[$8730$]3)2/2 = 6 см2. Высоту определим, рассмотрев треугольник AOA1. Так как основание ABCD - квадрат, то точка пересечения диагоналей O делит каждую из них пополам, то есть AO = OC = [$8730$]3 см. Сторона A1O наклонена под углом 60[$186$] к плоскости DBB1, которая в свою очередь перпендикулярна основанию, поэтому угол AOA1 между A1O и стороной AO, лежащей в плоскости основания, будет равен 90[$186$]-60[$186$]=30[$186$]. Тогда AA1 = AO[$183$]tg AOA1 = [$8730$]3[$183$]1/[$8730$]3 = 1 см - высота призмы, а её объём будет равен 6[$183$]1 = 6 см3.
Объём призмы равен произведению площади основания на высоту. В данном случае основанием правильной четырёхугольной призмы является квадрат с диагональю 2[$8730$]3 см, поэтому площадь основания будет равна (2[$8730$]3)2/2 = 6 см2. Высоту определим, рассмотрев треугольник AOA1. Так как основание ABCD - квадрат, то точка пересечения диагоналей O делит каждую из них пополам, то есть AO = OC = [$8730$]3 см. Сторона A1O наклонена под углом 60[$186$] к плоскости DBB1, которая в свою очередь перпендикулярна основанию, поэтому угол AOA1 между A1O и стороной AO, лежащей в плоскости основания, будет равен 90[$186$]-60[$186$]=30[$186$]. Тогда AA1 = AO[$183$]tg AOA1 = [$8730$]3[$183$]1/[$8730$]3 = 1 см - высота призмы, а её объём будет равен 6[$183$]1 = 6 см3.
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