The volume of a sphere is 72cm^3. What is the volume of a cone if its radius is the same as the sphere’s and its height is equal to the sphere’s diameter?
First, let's determine the radius and diameter of the sphere.
The formula for the volume of a sphere is V = (4/3)πr^3, where r is the radius of the sphere.
Given that the volume of the sphere is 72cm^3, we can solve for the radius:
72 = (4/3)πr^3
72 = (4/3)πr^3
r^3 = 72 / [(4/3)π]
r^3 = 54 / π
r = (54 / π)^(1/3) ≈ 3.48 cm
Now, let's calculate the height of the cone, which is equal to the sphere's diameter:
Height = 2 * radius ≈ 2 * 3.48 ≈ 6.96 cm
The formula for the volume of a cone is V = (1/3)πr^2h, where r is the radius of the cone and h is the height.
Plugging in the radius and height values, we get:
V = (1/3)π * (3.48)^2 * 6.96
V ≈ (1/3)π * 12.1104 * 6.96
V ≈ (1/3)π * 84.4656
V ≈ 88.919 cm^3
Therefore, the volume of the cone with the same radius as the sphere and a height equal to the sphere's diameter is approximately 88.919 cm^3.