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Чтобы найти объём описанного шара около правильной усеченной треугольной пирамиды, мы можем использовать следующую формулу:
V = (4/3) * π * R^3,
где V - объём шара, π - математическая константа (приближенное значение 3.14159), R - радиус описанной окружности вокруг основания усеченной пирамиды.
Для того чтобы найти радиус R, нам понадобятся высота пирамиды и размеры её оснований.
По условию, высота пирамиды равна 12 см, а стороны оснований соответственно равны √3 см и 7√3 см.
Обозначим a и b стороны оснований пирамиды. Тогда радиус R можно найти, используя следующую формулу:
R = (a * b * h) / (sqrt((a^2 + b^2 + ab)/3) + sqrt((a^2 + b^2 - ab)/3) + sqrt((a^2 - b^2 + ab)/3)),
где h - высота пирамиды.
Подставляя значения из условия, получим:
R = (√3 см * 7√3 см * 12 см) / (sqrt((√3^2 + 7√3^2 + √3 * 7√3)/3) + sqrt((√3^2 + 7√3^2 - √3 * 7√3)/3) + sqrt((√3^2 - 7√3^2 + √3 * 7√3)/3)).
Выполняя вычисления, получим:
R = (36√3 см^3) / (sqrt(3 + 63 + 21) + sqrt(3 + 63 - 21) + sqrt(3 - 63 + 21)).
R = (36√3 см^3) / (sqrt(87) + sqrt(45) + sqrt(-39)).
В данном случае, значение под корнем sqrt(-39) отрицательно, что означает, что радиус R является комплексным числом. Объём описанного шара, если радиус комплексный, не может быть определён в рамках классической геометрии.