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)%0D%0AResponses%0D%0A%0D%0A256 π cm3%0D%0AStart Fraction 25 over 6 End Fraction pi cm cubed%0D%0A%0D%0A25 π cm3%0D%0A25 pi cm cubed%0D%0A%0D%0A503 π cm3%0D%0AStart Fraction 50 over 3 End Fraction pi cm cubed%0D%0A%0D%0A252 π cm3
To solve this problem, we can use the formulas for the volume of a cone and the volume of a sphere.
The cone has a volume of 253 π cm^3, so we can set up the equation:
(1/3)πr^2h = 253π
where r is the radius and h is the height of the cone.
Since the height of the cone is equal to the sphere's diameter, we can say that h = 2r.
Substituting this into the equation, we get:
(1/3)πr^2(2r) = 253π
Simplifying the equation, we get:
(2/3)πr^3 = 253π
Dividing both sides by (2/3)π, we get:
r^3 = 253(3/2)
r^3 = 379.5
Taking the cube root of both sides, we find that r ≈ 7.44 cm.
Now, let's calculate the volume of the sphere using the formula:
V = (4/3)πr^3
V = (4/3)π(7.44^3)
V ≈ 4/3 * 3.14 * 388.98
V ≈ 1624.19π
Therefore, the volume of the sphere is approximately 1624.19 π cm^3.