The volume of a cone is 25/3 π cm^3 . What is the volume of a sphere if its radius is the same as the cone’s and the height of the cone is equal to the sphere’s diameter?

Answer 1

First, we need to find the radius and height of the cone.

The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius of the cone and h is the height of the cone.

Since the volume of the cone is 25/3 π cm^3, we have:

25/3 π = (1/3)πr^2h
25 = r^2h

Given that the height of the cone is equal to the diameter of the sphere, we have h = 2r.

Substitute h = 2r into the equation 25 = r^2h:

25 = r^2(2r)
25 = 2r^3

Solving for r, we get:

r^3 = 25/2
r = (25/2)^(1/3)

Now that we have the radius of the cone, we can calculate the volume of the sphere.

The volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius of the sphere.

Substitute r = (25/2)^(1/3) into the equation:

V = (4/3)π((25/2)^(1/3))^3
V = (4/3)π(125/27)
V = (500/81)π cm^3

Therefore, the volume of the sphere is (500/81)π cm^3.