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?
A. 25/6 π cm^3
B. 25/2 π cm^3
C.25 π cm^3
D. 50/3 π cm^3
First, let's find the height and radius of the cone.
Since the volume of the cone is given by V = 1/3 πr^2h, where r is the radius and h is the height.
25/3π = 1/3 πr^2h
r^2h = 25
Since the height of the cone is equal to the diameter of the sphere, we have h = 2r
r^2 *2r = 25
2r^3 = 25
r = 5/∛2
Now, we find the volume of the sphere using the formula V = 4/3 πr^3.
V = 4/3 π * (5/∛2)^3
V = 4/3 π * 125/∛8
V = 500/3 π / 2∛2
V = 250/3 π / ∛2
Therefore, the volume of the sphere is 250/3 π / ∛2 or 25/6 π cm^3, which is option A.