Question

В основании прямого параллелепипеда лежит ромб со стороной 12см и углом 30. меньшая диагональ параллелепипеда равна 13 см. найдите площадь полной поверхности параллелепипеда и объем.

Answer 1

Ответ:

Пошаговое объяснение:

Думаю 34

Answer 2

#### Problem Analysis We are given a parallelepiped (a three-dimensional figure with six faces) with a rhombus as its base. The rhombus has a side length of 12 cm and an angle of 30 degrees. The smaller diagonal of the parallelepiped is 13 cm. We need to find the total surface area and volume of the parallelepiped. #### Solution To find the total surface area of the parallelepiped, we need to calculate the areas of all six faces and then sum them up. 1. The area of the rhombus base can be calculated using the formula: **Area = (d1 * d2) / 2**, where d1 and d2 are the lengths of the diagonals of the rhombus. In this case, the smaller diagonal of the parallelepiped is equal to the longer diagonal of the rhombus. So, the area of the rhombus base is **Area = (13 * 12) / 2 = 78 cm^2**. 2. The other four faces of the parallelepiped are rectangles. The lengths of these rectangles are equal to the side length of the rhombus (12 cm), and their widths can be calculated using trigonometry. Since we know the angle of the rhombus is 30 degrees, we can use the sine function to find the width of the rectangles. The formula for the width of the rectangles is **Width = side length * sin(angle)**. - The width of the rectangle opposite to the rhombus base is **Width = 12 * sin(30) = 6 cm**. - The width of the rectangle adjacent to the rhombus base is also **6 cm**. 3. The remaining two faces of the parallelepiped are parallelograms. The lengths of these parallelograms are equal to the side length of the rhombus (12 cm), and their heights can be calculated using trigonometry. Since we know the angle of the rhombus is 30 degrees, we can use the cosine function to find the height of the parallelograms. The formula for the height of the parallelograms is **Height = side length * cos(angle)**. - The height of the parallelogram opposite to the rhombus base is **Height = 12 * cos(30) = 10.392 cm**. - The height of the parallelogram adjacent to the rhombus base is also **10.392 cm**. 4. Now, we can calculate the areas of the remaining four faces of the parallelepiped. The area of a rectangle is **Area = length * width**, and the area of a parallelogram is **Area = base * height**. - The area of the rectangle opposite to the rhombus base is **Area = 12 * 6 = 72 cm^2**. - The area of the rectangle adjacent to the rhombus base is also **72 cm^2**. - The area of the parallelogram opposite to the rhombus base is **Area = 12 * 10.392 = 124.704 cm^2**. - The area of the parallelogram adjacent to the rhombus base is also **124.704 cm^2**. 5. Finally, we can calculate the total surface area of the parallelepiped by summing up the areas of all six faces. - Total surface area = Area of the rhombus base + 2 * Area of the rectangle + 2 * Area of the parallelogram - Total surface area = 78 + 2 * 72 + 2 * 124.704 - Total surface area = 78 + 144 + 249.408 - Total surface area = 471.408 cm^2. 6. To find the volume of the parallelepiped, we can use the formula: **Volume = base area * height**, where the base area is the area of the rhombus base and the height is the length of the smaller diagonal of the parallelepiped. - Volume = 78 * 13 - Volume = 1014 cm^3. #### Answer The total surface area of the parallelepiped is **471.408 cm^2** and the volume is **1014 cm^3**.