Question

Высота конуса равна 5 см, а угол при вершине осевого сечения равен 120 градусов. Найдите объём конуса. С рисунком пожалуйста!

Answer 1

Решение данной задачи

Answer 2

Calculation of the Cone Volume

To find the volume of a cone, we need to know the height of the cone and the angle at the vertex of the axial section. In this case, the height of the cone is given as 5 cm, and the angle at the vertex of the axial section is 120 degrees.

The formula to calculate the volume of a cone is:

V = (1/3) * π * r^2 * h

Where: - V is the volume of the cone - π is a mathematical constant approximately equal to 3.14159 - r is the radius of the base of the cone - h is the height of the cone

To find the radius of the base of the cone, we need to use trigonometry. The angle at the vertex of the axial section is 120 degrees, which means that the angle at the center of the base of the cone is half of that, or 60 degrees.

Using trigonometry, we can find the radius of the base of the cone using the formula:

r = h * tan(θ)

Where: - r is the radius of the base of the cone - h is the height of the cone - θ is the angle at the center of the base of the cone

Let's calculate the volume of the cone using the given values:

Step 1: Convert the angle from degrees to radians: - 120 degrees = 120 * (π/180) radians = 2.094 radians

Step 2: Calculate the radius of the base of the cone: - r = 5 cm * tan(60 degrees) = 5 cm * tan(π/3 radians) = 5 cm * √3 ≈ 8.660 cm

Step 3: Calculate the volume of the cone: - V = (1/3) * π * (8.660 cm)^2 * 5 cm ≈ 150.796 cm^3

Therefore, the volume of the cone is approximately 150.796 cm^3.

Diagram of the Cone

Unfortunately, I am unable to provide a diagram in this text-based format. However, you can visualize a cone as a three-dimensional shape with a circular base and a pointed top. The height of the cone is the distance from the base to the top, and the radius of the base is the distance from the center of the base to any point on the edge of the base.

I hope this helps! Let me know if you have any further questions.