Question

Площядь боковой поверхности конуса ровна 20П кв.см а площядь его основания на 4П кв.см меньше Найти обьем конуса

Answer 1

S(бок)=20П(кв см)

S(осн)=20П-4П=16П(кв см)

 

S(бок)=ПRl=20П

             Rl=20, l=20/R

S(осн)=ПR^2=16П

             R^2=16

             R=4(см)-радиус конуса

l=20/R=20/4=5(см)-образующая конуса

Высота конуса h=sqrt{l^2-R^2}=sqrt{5^2-4^2}=3 (см)

Объём конуса V=1/3 *S(осн)*h

V=1/3 *16П * 3=16П (см куб)

 

Answer 2

Calculation of Cone Volume

To find the volume of a cone, we need to know the area of its base and the height of the cone. In this case, we are given the information that the lateral surface area of the cone is 20π square centimeters and the area of its base is 4π square centimeters less than that.

Let's denote the area of the base as A and the height of the cone as h.

We know that the lateral surface area of a cone is given by the formula:

Lateral Surface Area = π * r * l

where r is the radius of the base and l is the slant height of the cone.

The slant height can be calculated using the Pythagorean theorem:

l = √(r^2 + h^2)

Since the lateral surface area is given as 20π square centimeters, we can write the equation:

20π = π * r * √(r^2 + h^2)

Simplifying the equation, we get:

20 = r * √(r^2 + h^2)

Now, we are also given that the area of the base is 4π square centimeters less than the lateral surface area. So we can write:

A = 20π - 4π = 16π

The area of the base of a cone is given by the formula:

Base Area = π * r^2

So we have:

16π = π * r^2

Simplifying the equation, we get:

16 = r^2

Taking the square root of both sides, we find:

r = 4

Now, we can substitute the value of r into the equation for the lateral surface area:

20 = 4 * √(4^2 + h^2)

Simplifying the equation, we get:

5 = √(16 + h^2)

Squaring both sides of the equation, we find:

25 = 16 + h^2

Subtracting 16 from both sides, we get:

h^2 = 9

Taking the square root of both sides, we find:

h = 3

Now that we have the values of r and h, we can calculate the volume of the cone using the formula:

Volume = (1/3) * π * r^2 * h

Substituting the values, we get:

Volume = (1/3) * π * 4^2 * 3 = 16π

Therefore, the volume of the cone is 16π cubic centimeters.