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Консультация № 197690
04.02.2020, 22:49
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Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
Точка О - середина ребра А1В1 прямоугольного параллелепипеда АВСDA1B1C1D1, основанием которого является квадрат. Боковое ребро параллелепипеда в три раза больше ребра основания. Сумма площадей треугольников В1ОВ и А1ОD1 равна 25см2. Вычислите объём параллелепипеда ABCDA1B1C1D1.
Точка О - середина ребра А1В1 прямоугольного параллелепипеда АВСDA1B1C1D1, основанием которого является квадрат. Боковое ребро параллелепипеда в три раза больше ребра основания. Сумма площадей треугольников В1ОВ и А1ОD1 равна 25см2. Вычислите объём параллелепипеда ABCDA1B1C1D1.
Обсуждение
09.02.2020, 19:06
общий
это ответ
Здравствуйте, svrvsvrv!
Поскольку точка О - середина ребра А1В1, принадлежащего прямоугольнику ABB1A1 (одной из боковых граней) и квадрату A1B1C1D1 (верхнему основанию), то треугольники В1ОВ и А1ОD1 составят одну четверть от ABB1A1 и A1B1C1D1 соответственно. Тогда сумма площадей этих двух граней будет в четыре раза больше суммы площадей треугольников и составит 4[$183$]25 = 100 см2. Поскольку боковое ребро в три раза больше ребра основания, то и площади боковой грани и основания будут различаться в 3 раза и равняться 75 см2 и 25 см2 соответственно (что как раз в сумме даёт 100 см2). Поскольку основание - квадрат, то его ребро равно [$8730$]25 = 5 см, боковое ребро - 3[$183$]5 = 15 см (высота параллелепипеда), а объём параллелепипеда составит 25[$183$]15 = 375 см3.
Поскольку точка О - середина ребра А1В1, принадлежащего прямоугольнику ABB1A1 (одной из боковых граней) и квадрату A1B1C1D1 (верхнему основанию), то треугольники В1ОВ и А1ОD1 составят одну четверть от ABB1A1 и A1B1C1D1 соответственно. Тогда сумма площадей этих двух граней будет в четыре раза больше суммы площадей треугольников и составит 4[$183$]25 = 100 см2. Поскольку боковое ребро в три раза больше ребра основания, то и площади боковой грани и основания будут различаться в 3 раза и равняться 75 см2 и 25 см2 соответственно (что как раз в сумме даёт 100 см2). Поскольку основание - квадрат, то его ребро равно [$8730$]25 = 5 см, боковое ребро - 3[$183$]5 = 15 см (высота параллелепипеда), а объём параллелепипеда составит 25[$183$]15 = 375 см3.
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