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)
Let's first find the radius and height of the cone.
The volume of a cone is given by the formula:
Vcone = (1/3)πr^2h
Given that Vcone = 253π cm^3, we can substitute this value and the fact that the cone's height is equal to the sphere's diameter to get:
253π = (1/3)πr^2(d)
where d is the sphere's diameter and r is the radius of both the cone and the sphere. Since the diameter is twice the radius, we can simplify the equation to:
253π = (1/3)πr^2(2r)
253π = (2/3)πr^3
Now we can solve for r:
253π = (2/3)πr^3
253 = (2/3)r^3
r^3 = 379.5
r = (379.5)^(1/3)
r ≈ 7.56 cm
So, the radius of the cone and sphere is approximately 7.56 cm. Now, we can find the volume of the sphere using the formula for the volume of a sphere:
Vsphere = (4/3)πr^3
Vsphere = (4/3)π(7.56)^3
Vsphere ≈ 1432.5π cm^3
Therefore, the volume of the sphere is approximately 1432.5π cm^3.