Question

Площадь боковой поверхности конуса равно 20 пи см2, а его образующая имеет длину 5 см. Найдите объём конуса

Answer 1

S=ПRL R=S/ПL=20/5П=4/П

V=1/3ПR^2h

h^2=25-16/П^2

V=1/3П16/П^2sqrt((П^2*25-16))/П=16sqrt((П^2*25-16)/3П^2

 

Answer 2

Calculation of Cone Volume

To find the volume of a cone, we need to know the formula for calculating it. The formula for 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

In this case, we are given the area of the lateral surface of the cone and the length of the slant height (or the generatrix). However, we need the radius and height to calculate the volume. We can use the given information to find the radius and height.

Finding the Radius and Height

The area of the lateral surface of the cone is given as 20π cm^2, and the length of the generatrix is given as 5 cm.

Let's denote the radius of the base of the cone as r and the height of the cone as h.

The formula for the lateral surface area of a cone is:

S = π * r * L

Where: - S is the lateral surface area of the cone - L is the length of the generatrix

From the given information, we have:

20π = π * r * 5

Simplifying the equation, we find:

r = 4 cm

Now, we can use the Pythagorean theorem to find the height of the cone. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In this case, the height of the cone is the perpendicular side, the radius is the base side, and the generatrix is the hypotenuse.

Using the Pythagorean theorem, we have:

h^2 = L^2 - r^2

Substituting the given values, we get:

h^2 = 5^2 - 4^2

Simplifying the equation, we find:

h = 3 cm

Calculation of Cone Volume

Now that we have the radius and height of the cone, we can calculate its volume using the formula mentioned earlier:

V = (1/3) * π * r^2 * h

Substituting the values we found, we get:

V = (1/3) * π * 4^2 * 3

Simplifying the equation, we find:

V = 16π cm^3

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

Please note that the values of the radius and height were calculated based on the given information, and the volume was calculated using the formula for the volume of a cone.