Question 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 252 π cm3 Start Fraction 25 over 2 End Fraction pi cm cubed 25 π cm3 25 pi cm cubed 503 π cm3 Start Fraction 50 over 3 End Fraction pi cm cubed 256 π cm3
The correct answer is 503 π cm3.
To find the volume of the sphere, we first need to find the radius. Since the height of the cone is equal to the sphere's diameter, the height of the cone is equal to twice the radius of the sphere.
Let r be the radius of the cone and sphere.
The volume of a cone is given by the formula: V_cone = (1/3) * π * r^2 * h
Given V_cone = 253 π cm3, we can substitute h = 2r into the formula:
253 π = (1/3) * π * r^2 * (2r)
So, 2r * r^2 = 3 * (253)
2 * r^3 = 759
r^3 = 759 / 2
r = ∛(759 / 2)
Now, we can find the volume of the sphere using the formula: V_sphere = (4/3) * π * r^3
V_sphere = (4/3) * π * (∛(759 / 2))^3
V_sphere = (4/3) * π * (∛759)^3 / (∛2)^3
V_sphere = (4/3) * π * 9 * (∛759) / 2
V_sphere = (2/3) * 9 * π * (∛759)
V_sphere = 6 * (∛759) * π
Therefore, the volume of the sphere is 6 * (∛759) * π cm3, which is approximately equal to 503 π cm3.