Question

Основанием прямой четырехугольной призмы является ромб с диагоналями 12 см и 16 см. Найдите полную поверхность призмы, если больше диагональ призмы наклонена к основанию под углом 45°.

Answer 1

РЕШЕНИЕ
Площадь основания по формуле
Sосн = d₁*d₂/2 = 12*16/2 = 96 см² - площадь основания
h = d₂ = 16 - высота при угле диагонали 45°
Сторона ромба по теореме Пифагора, где
a = d₁/2 = 6 см  - катет
b = d₂/2 = 8 см - катет
с² = a²+b² = 36 + 64 = 100
c = √100 = 10 - сторона ромба.
Sбок = P*h = 4*a*h = 40*16 = 640 см² - боковая поверхность.
Sполн = 640 + 2*96 = 640+192 = 832 см² - полная поверхность - ОТВЕТ

Answer 2

Calculation of the Total Surface Area of the Prism

To find the total surface area of the prism, we need to calculate the areas of all its faces and then sum them up.

The given prism has a rhombus as its base, with diagonals measuring 12 cm and 16 cm. The larger diagonal of the prism is inclined to the base at an angle of 45°.

Let's break down the calculation step by step:

1. Calculate the area of the rhombus base: - The area of a rhombus can be calculated using the formula: Area = (d1 * d2) / 2, where d1 and d2 are the lengths of the diagonals. - In this case, the diagonals are 12 cm and 16 cm. - Plugging in the values, we get: Area = (12 cm * 16 cm) / 2 = 96 cm². 2. Calculate the area of the inclined face: - The inclined face is a parallelogram with one side equal to the length of the larger diagonal (16 cm) and the height equal to the length of the smaller diagonal (12 cm). - The area of a parallelogram can be calculated using the formula: Area = base * height. - Plugging in the values, we get: Area = 16 cm * 12 cm = 192 cm².

3. Calculate the areas of the other four faces: - Since the prism is a right prism, the areas of the other four faces are equal to the area of the base (96 cm²).

4. Calculate the total surface area: - The total surface area of the prism is the sum of the areas of all its faces. - Adding up the areas, we get: Total Surface Area = 96 cm² + 192 cm² + 96 cm² + 96 cm² + 96 cm² = 576 cm².

Therefore, the total surface area of the given prism is 576 cm².

Please note that the calculations provided above are based on the given information and assumptions about the shape and dimensions of the prism.