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 π CM^3
B: 25/2 π CM^3
C: 50/3 π CM^3
D: 25/6 π CM^3
B: 25/2 π cm^3
The volume of a cone is given by V = 1/3 π r^2 h, where r is the radius and h is the height.
Given that the volume of the cone is 25/3 π cm^3, we can rearrange the formula to solve for h: h = (3V) / (π r^2).
Substitute the given values: h = (3(25/3 π)) / (π r^2) = 25 / r
Since the height of the cone is equal to the diameter of the sphere, the height of the cone is equal to 2r, where r is the radius of the cone.
Therefore, h = 2r = 25 / r
Solve for r: r^2 = 25, r = 5
The volume of a sphere is given by V = 4/3 π r^3
Substitute the radius of the cone, r = 5, into the formula: V = 4/3 π (5)^3 = 4/3 π 125 = 500/3 π = 166 2/3 π
Therefore, the volume of the sphere is 25/2 π cm^3.