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?
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.