Question

Образующая прямого конуса равна 4 см и наклонена к плоскости основания под углом 30 градусов найдите объем конуса

Answer 1

Cos30=r/l
r=l * cos30. h=l*sin30=2
r=4*(корень из 3)/2=2кор из 3
Sосн=12n
V=(12n*2)/3=8n

Answer 2

Calculation of Cone Volume

To find the volume of a cone, we need to know the height and the radius of the base. In this case, we are given the slant height and the angle of inclination to the base plane. We can use this information to calculate the height and the radius of the base, and then use the formula for the volume of a cone.

Let's break down the problem step by step:

1. Given information: - Slant height of the cone = 4 cm - Angle of inclination to the base plane = 30 degrees

2. Calculating the height and the radius of the base: - We can use trigonometry to find the height and the radius of the base. - The slant height, height, and radius form a right triangle. - The slant height is the hypotenuse, the height is the opposite side, and the radius is the adjacent side. - We can use the sine and cosine functions to calculate the height and the radius.

- Let's denote the height as h and the radius as r. - Using trigonometry, we have: - sin(30 degrees) = h / 4 cm - cos(30 degrees) = r / 4 cm

- Solving these equations, we can find the values of h and r.

3. Calculating the volume of the cone: - The formula for the volume of a cone is V = (1/3) * π * r^2 * h. - Once we have the values of h and r, we can substitute them into the formula to calculate the volume.

Let's calculate the height and the radius first:

Using the given angle of inclination of 30 degrees, we can find the height and the radius of the base using trigonometry.

sin(30 degrees) = h / 4 cm cos(30 degrees) = r / 4 cm

Solving these equations, we find:

h = 4 cm * sin(30 degrees) = 2 cmr = 4 cm * cos(30 degrees) = 3.464 cm Now that we have the height and the radius, we can calculate the volume of the cone using the formula:

V = (1/3) * π * r^2 * h

Substituting the values:

V = (1/3) * π * (3.464 cm)^2 * 2 cm

Calculating this expression, we find:

V ≈ 20.786 cm^3

Therefore, the volume of the cone is approximately 20.786 cm^3.

Please note that the calculations are based on the given information and assumptions made.