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 256 π cm3 Start Fraction 25 over 6 End Fraction pi cm cubed 252 π cm3 Start Fraction 25 over 2 End Fraction pi cm cubed 503 π cm3 Start Fraction 50 over 3 End Fraction pi cm cubed 25 π cm3 25 pi cm cubed
The correct answer is 503 π cm3.
Here's how we can calculate it:
The volume of a cone is given by the formula:
Vcone = (1/3)πr^2h
Given that the volume of the cone is 253 π cm3, and the height of the cone is equal to the sphere's diameter, we can write:
253 π = (1/3)πr^2h
Since the height of the cone is equal to the sphere's diameter, we have:
h = 2r
Substituting this into the equation above:
253 π = (1/3)πr^2(2r)
Simplifying:
253 = (2/3)r^3
To find the radius of the cone, we need to solve for r:
r^3 = (3/2) * 253
r^3 = 379.5
Taking the cube root of both sides:
r ≈ 7.33
Now, we can calculate the volume of the sphere using the formula:
Vsphere = (4/3)πr^3
Substituting in the value of r:
Vsphere = (4/3)π(7.33)^3
≈ 503 π cm3
Therefore, the volume of the sphere is approximately 503 π cm3.