Question

Радиус основания цилиндра 4 см, площадь боковой поверхности вдвое больше площади основания. Найти объем цилиндра.

Answer 1

Sоснования=πr²=3.14*4²=3.14*16=50.24см²
Sбоковой поверхности=50,24*2=100,48см²
Lоснования=2πr=2*3.14*4=8*3.14=25.12 см
Hцилиндра=100,48:25,12=4 см
V=Sоснования*Hцилиндра=50,24*4=200,96 см³

Answer 2

Problem Analysis

We are given that the radius of the base of a cylinder is 4 cm, and the lateral surface area is twice the area of the base. We need to find the volume of the cylinder.

Solution

To find the volume of a cylinder, we need to know the radius of the base and the height of the cylinder. However, the height is not given in the problem statement. We can solve for the height using the given information.

Let's assume the height of the cylinder is h.

The formula for the lateral surface area of a cylinder is given by: Lateral Surface Area = 2 * π * r * h, where r is the radius of the base and h is the height of the cylinder.

According to the problem statement, the lateral surface area is twice the area of the base. The formula for the area of the base of a cylinder is given by: Base Area = π * r^2.

Therefore, we can write the equation as: 2 * π * r * h = 2 * π * r^2.

Simplifying the equation, we get: h = r.

Now that we know the height of the cylinder is equal to the radius of the base, we can proceed to find the volume.

The formula for the volume of a cylinder is given by: Volume = Base Area * Height.

Substituting the values, we get: Volume = π * r^2 * h.

Since h = r, we can simplify the equation to: Volume = π * r^2 * r.

Finally, we can calculate the volume of the cylinder using the given radius.

Calculation

Given that the radius of the base of the cylinder is 4 cm, we can substitute this value into the formula for the volume of a cylinder.

Volume = π * (4 cm)^2 * 4 cm.

Simplifying the equation, we get: Volume = 16π cm^3.

Therefore, the volume of the cylinder is 16π cm^3.

Answer

The volume of the cylinder is 16π cm^3.

Verification

Let's verify the answer using the given information.

The lateral surface area of the cylinder is twice the area of the base. The formula for the lateral surface area is given by: Lateral Surface Area = 2 * π * r * h.

Substituting the values, we get: Lateral Surface Area = 2 * π * 4 cm * 4 cm.

Simplifying the equation, we get: Lateral Surface Area = 32π cm^2.

The area of the base of the cylinder is given by: Base Area = π * r^2.

Substituting the values, we get: Base Area = π * (4 cm)^2.

Simplifying the equation, we get: Base Area = 16π cm^2.

Since the lateral surface area is twice the area of the base, we have: Lateral Surface Area = 2 * Base Area.

Substituting the values, we get: 32π cm^2 = 2 * 16π cm^2.

Simplifying the equation, we get: 32π cm^2 = 32π cm^2.

The equation is true, which verifies that the volume of the cylinder is calculated correctly.

Conclusion

The volume of the cylinder with a radius of 4 cm is 16π cm^3.