The volume of a sphere is 48 m^3 . What is the volume of a cylinder if its radius is the same as the sphere’s and its height is equal to the sphere’s diameter?
The formula for the volume of a sphere is V = (4/3)πr^3, where V is the volume and r is the radius.
We're given that the volume of the sphere is 48 m^3, so:
48 = (4/3)πr^3
To find the radius, we need to rearrange this equation:
r^3 = (3/4)(48/π)
r^3 = 36/π
r ≈ (36/π)^(1/3)
The volume of a cylinder is given by V = πr^2h, where V is the volume, r is the radius, and h is the height. If the radius of the cylinder is the same as the sphere's and the height is equal to the sphere's diameter, then the height is 2r.
Substituting the value of r we found into the volume formula for the cylinder, we get:
V = π[(36/π)^(1/3)]^2 * 2[(36/π)^(1/3)]
V = π * (36/π)^(2/3) * 2(36/π)^(1/3)
Simplifying further,
V = 2π * (36/π) * (36/π)^(1/3)
V = 2 * 36^(4/3) * π^(2/3)
V ≈ 2 * 4.876 * π^(2/3)
V ≈ 9.752 * π^(2/3)
Therefore, the volume of the cylinder is approximately 9.752 * π^(2/3) cubic meters.