Question

Прямоугольник со сторонами 11 см и 6 см вращается вокруг прямой, проходящей через середины меньших сторон. Найти обьем полученного тела

Answer 1

Объём цилиндра равен
D=6см
h=11см
пи/4=0,785

V=0.785*D*D*h=0.785*6*6*11=310.86 см.куб.

Answer 2

Calculation of the Volume of the Solid

To find the volume of the solid obtained by rotating a rectangle with sides measuring 11 cm and 6 cm around a line passing through the midpoints of the shorter sides, we can use the method of cylindrical shells.

The volume of the solid can be calculated by integrating the area of the cylindrical shells formed by rotating the rectangle.

Let's denote the length of the longer side of the rectangle as L (11 cm) and the length of the shorter side as W (6 cm).

The line passing through the midpoints of the shorter sides divides the rectangle into two equal parts. Let's call the distance between this line and the longer side of the rectangle as h.

To find the value of h, we can use the Pythagorean theorem. The hypotenuse of the right triangle formed by h, W/2, and L/2 is equal to the length of the longer side of the rectangle.

Using the Pythagorean theorem, we can calculate h as follows:

h = sqrt((L/2)^2 - (W/2)^2)

Substituting the values of L and W into the equation, we get:

h = sqrt((11/2)^2 - (6/2)^2) = sqrt(121/4 - 9/4) = sqrt(112/4) = sqrt(28) = 2sqrt(7) cm

Now, we can calculate the volume of the solid using the formula for the volume of a cylindrical shell:

V = 2πh * A

where π is the mathematical constant pi (approximately 3.14159), h is the height of the cylindrical shell, and A is the area of the cylindrical shell.

The height of each cylindrical shell is equal to the width of the rectangle, which is W (6 cm).

The area of each cylindrical shell can be calculated by multiplying the circumference of the shell (equal to the perimeter of the rectangle) by its height.

The perimeter of the rectangle is equal to 2L + 2W.

Substituting the values of L and W, we get:

Perimeter = 2(11) + 2(6) = 22 + 12 = 34 cm

Therefore, the area of each cylindrical shell is equal to 34 * 6 = 204 cm^2.

Finally, we can calculate the volume of the solid by substituting the values into the formula:

V = 2π(2sqrt(7)) * 204 = 408πsqrt(7) cm^3

So, the volume of the solid obtained by rotating the rectangle with sides measuring 11 cm and 6 cm around a line passing through the midpoints of the shorter sides is 408πsqrt(7) cm^3.

Please note that this calculation assumes that the rectangle is rotated 360 degrees around the line passing through the midpoints of the shorter sides.