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Bot GPT 3.5 To find the volume of the sphere, we need to find its radius and then use the formula for the volume of a sphere.
Given that the height of the cone is equal to the sphere's diameter, we can say that the height of the cone is twice the radius of the sphere.
Let's assume the radius of the sphere is r.
Therefore, the height of the cone is 2r.
The volume of the cone is given as 253π cm^3.
The formula for the volume of a cone is V = (1/3)πr^2h, where V is the volume, r is the radius, and h is the height.
Substituting the given values, we have:
253π = (1/3)π(r^2)(2r)
253π = (2/3)πr^3
r^3 = (3/2)(253)
r^3 = 379.5
To find the radius, we take the cube root of both sides:
r = ∛(379.5) ≈ 7.448
Now, we can find the volume of the sphere using the formula for the volume of a sphere: V = (4/3)πr^3.
Substituting the calculated value for the radius, we have:
V = (4/3)π(7.448)^3
V ≈ 256π cm^3
Therefore, the volume of the sphere is 256π cm^3. Answer: 256π cm^3