The volume of a cone is 25/3 cm³. 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. 50/3 pi cm³
B. 25/6 pi cm³
C. 25/2 pi cm³
D. 25 pi cm³
The formula for the volume of a cone is V = (1/3)πr²h, where r is the radius of the cone and h is the height of the cone.
Given that the volume of the cone is 25/3 cm³, we can substitute this value into the formula: 25/3 = (1/3)πr²h
Since the height of the cone is equal to the sphere's diameter, we can substitute h = 2r into the equation: 25/3 = (1/3)πr²(2r)
Simplifying the equation, we get: 25/3 = (2/3)πr³
Multiplying both sides by 3/2, we get: 25/2 = πr³
Now, the formula for the volume of a sphere is V = (4/3)πr³, where r is the radius of the sphere.
Comparing this formula to our equation above, we see that the volume of the sphere is also 25/2 cm³.
Therefore, the volume of the sphere is 25/2 pi cm³, which is option C.