Question

Разверткой боковой поверхности цилиндра служит прямоугольник, одна сторона которого в 2 раза больше другойплощадь боковой поверхности цилиндра =20 найдите площадь его поверхности

Answer 1

Х - одна сторона прямоугольника
2х - вторая сторона прямоугольника 
S бок = х *2х 
20 = 2х²
х² = 10
х = √10 
2х = 2√10  -эта сторона так же является длиной окружности цилиндра
L = 2√10
L = 2πR
R = L/2π
Sокр = πR² = π(L/2π)² = πL²/ 4π² = L²/4π
S окр = (2√10)²/4 π = 4*10/4π = 10/π
S бок = √10 * 2√10 = 2*10 = 20
Sпол = Sбок + 2Sокр
S пол = 20 + 2(10/π) = 20 +(20/π) = (20π +20)/π 

Answer 2

Finding the Length and Width of the Rectangle

To find the area of the lateral surface of a cylinder, we need to determine the dimensions of the rectangle that serves as the unwrapped lateral surface of the cylinder. The problem states that one side of the rectangle is twice as long as the other side.

Let's assume that the shorter side of the rectangle is represented by x. Therefore, the longer side of the rectangle would be 2x.

Calculating the Area of the Lateral Surface of the Cylinder

The area of the lateral surface of a cylinder can be calculated using the formula:

Area of Lateral Surface = Perimeter of the Base x Height

In this case, the base of the cylinder is a circle, and the height is the length of the rectangle.

The perimeter of the base of a cylinder is given by the formula:

Perimeter of the Base = 2πr

where r is the radius of the base.

Since the problem does not provide the radius or height of the cylinder, we cannot calculate the exact area of the lateral surface. However, we can still provide a general formula for the area based on the given information.

General Formula for the Area of the Lateral Surface

Let's assume that the radius of the base of the cylinder is represented by r and the height of the cylinder is represented by h.

The area of the lateral surface of the cylinder can be calculated using the formula:

Area of Lateral Surface = 2πrh

Substituting the values we found earlier for the dimensions of the rectangle, we can rewrite the formula as:

Area of Lateral Surface = 2πr(2x)

Simplifying further, we get:

Area of Lateral Surface = 4πrx

Since the problem states that the area of the lateral surface of the cylinder is equal to 20, we can set up the equation:

4πrx = 20

Solving for the Area of the Surface

To find the area of the surface, we need to know the values of r and x. Unfortunately, the problem does not provide these values, so we cannot calculate the exact area.

However, we can still provide a general formula for the area based on the given information. If we assume that r and x are positive real numbers, we can solve the equation for x:

4πrx = 20

Dividing both sides of the equation by 4πr, we get:

x = 20 / (4πr)

Substituting this value of x back into the formula for the area of the lateral surface, we get:

Area of Lateral Surface = 4πr(20 / (4πr))

Simplifying further, we get:

Area of Lateral Surface = 20

Therefore, based on the given information, the area of the lateral surface of the cylinder is 20 square units.