Question

. Прямоугольник со сторонами 10 см и 8 см вращают вокруг диагонали. Найдите объем полученного тела.

Answer 1

(10+10)+(8+8)=36(см.)

Answer 2

Problem Analysis

We are given a rectangle with sides measuring 10 cm and 8 cm. The rectangle is rotated around its diagonal, and we need to find the volume of the resulting solid.

Solution

To find the volume of the solid obtained by rotating the rectangle around its diagonal, we can use the formula for the volume of a frustum of a cone. The frustum of a cone is a solid obtained by cutting a cone with a plane parallel to its base.

The formula for the volume of a frustum of a cone is given by:

V = (1/3) * π * h * (r1^2 + r2^2 + r1 * r2)

Where: - V is the volume of the frustum - π is the mathematical constant pi (approximately 3.14159) - h is the height of the frustum - r1 and r2 are the radii of the two circular bases of the frustum

In our case, the rectangle is rotated around its diagonal, which is also the height of the frustum. The radii of the two circular bases of the frustum can be calculated using the lengths of the sides of the rectangle.

Let's calculate the volume of the frustum using the given dimensions.

Calculation

The rectangle has sides measuring 10 cm and 8 cm. The diagonal of the rectangle can be calculated using the Pythagorean theorem:

d = sqrt(10^2 + 8^2)

Let's calculate the value of d:

d = sqrt(100 + 64) = sqrt(164) ≈ 12.81 cm

The height of the frustum is equal to the diagonal of the rectangle, which is approximately 12.81 cm.

The radii of the two circular bases of the frustum can be calculated using the lengths of the sides of the rectangle:

r1 = 10/2 = 5 cm

r2 = 8/2 = 4 cm

Now, let's substitute the values into the formula for the volume of the frustum:

V = (1/3) * π * h * (r1^2 + r2^2 + r1 * r2)

V = (1/3) * π * 12.81 * (5^2 + 4^2 + 5 * 4)

V ≈ 1/3 * 3.14159 * 12.81 * (25 + 16 + 20)

V ≈ 1/3 * 3.14159 * 12.81 * 61

V ≈ 1/3 * 3.14159 * 12.81 * 61 ≈ 1/3 * 3.14159 * 780.41 ≈ 820.68 cm^3

Therefore, the volume of the solid obtained by rotating the rectangle around its diagonal is approximately 820.68 cm^3.

Answer

The volume of the solid obtained by rotating the rectangle with sides measuring 10 cm and 8 cm around its diagonal is approximately 820.68 cm^3.