The volume of a cone is 25/3 π 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?
First, we need to find the radius of the cone. The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height.
Given that the volume of the cone is 25/3 π cm^3 and the height of the cone is equal to the sphere’s diameter, we can rewrite the formula as:
25/3 π = (1/3)πr^2(2r)
25/3 = 2r^3
r^3 = 25/6
r = (25/6)^(1/3) = (25/216)^(1/3)
Now, the volume of a sphere is given by the formula V = (4/3)πr^3. Since the radius of the sphere is the same as the cone’s, the radius of the sphere is also (25/216)^(1/3). Substituting this into the formula for the volume of the sphere, we get:
V = (4/3)π((25/216)^(1/3))^3
= (4/3)π(25/216)
= (100π/3)(25/216)
= 2500π/648
≈ 12.19 cm^3
Therefore, the volume of the sphere is approximately 12.19 cm^3.