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
503 π cm3
Start Fraction 50 over 3 End Fraction pi cm cubed
25 π cm3
25 pi cm cubed
252 π cm3
Start Fraction 25 over 2 End Fraction pi cm cubed
256 π cm3
The volume of a cone is given by the formula Vcone = (1/3)πr^2h, where r is the radius and h is the height of the cone.
Given that the volume of the cone is 253π cm^3, and the height of the cone is equal to the sphere's diameter, we can find the radius of the cone.
Let's assume the radius of the cone is r. Then the height of the cone is 2r (since it is equal to the sphere's diameter).
So, substituting the values into the formula, we have:
253π = (1/3)πr^2(2r)
Simplifying, we get:
3(253) = 2r^3
759 = 2r^3
Dividing both sides by 2, we have:
379.5 = r^3
Taking the cube root of both sides, we find:
r ≈ 7.71
Now, the volume of a sphere is given by the formula Vsphere = (4/3)πr^3, where r is the radius of the sphere.
Substituting the value of r that we just found, we have:
Vsphere = (4/3)π(7.71)^3
Vsphere ≈ 503.46π
Therefore, the volume of the sphere is approximately 503.46π cm^3.