The volume of a cone is 253 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?(1 point) Responses 25 π cm3 25 pi cm cubed 503 π cm3 Start Fraction 50 over 3 End Fraction pi cm cubed 252 π cm3 Start Fraction 25 over 2 End Fraction pi cm cubed 256 π cm3 Start Fraction 25 over 6 End Fraction pi cm cubed
First, let's find the radius of the cone using the formula for the volume of a cone:
V = 1/3 * π * r^2 * h
where V is the volume, r is the radius, and h is the height.
Since the volume of the cone is given as 253π cm^3 and the height of the cone is equal to the sphere's diameter, we have:
253π = 1/3 * π * r^2 * (2r)
Simplifying this equation, we get:
253 = 2/3 * r^3
Multiplying both sides by 3/2:
r^3 = 253 * (3/2)
r^3 = 379.5
Taking the cube root of both sides, we find:
r ≈ 7.25
Now, let's find the volume of the sphere using the formula for the volume of a sphere:
V = 4/3 * π * r^3
Substituting the radius of the sphere, which is the same as the radius of the cone, we have:
V = 4/3 * π * (7.25)^3
V ≈ 4/3 * π * 378.265625
V ≈ 503.02π
Therefore, the volume of the sphere is approximately 503π cm^3.