The volume of a sphere is 48 m'. 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 volume of a sphere is given by the formula V = (4/3)πr^3 , where V is the volume and r is the radius.
Given that the volume of the sphere is 48 m³, we can set up the equation as follows:
48 = (4/3)πr^3
To solve for r, we can rearrange the equation:
r^3 = (3/4)(48/π)
r^3 = 36/π
r = (36/π)^(1/3)
Now, since the radius of the cylinder is the same as the sphere's, the radius of the cylinder is also (36/π)^(1/3).
The height of the cylinder is equal to the sphere's diameter, which is twice the sphere's radius:
height = 2 * (36/π)^(1/3)
The volume of a cylinder is given by the formula V = πr^2h , where V is the volume, r is the radius, and h is the height.
Therefore, the volume of the cylinder is:
V = π * (36/π)^(1/3)^2 * 2 * (36/π)^(1/3)
Simplifying, we get:
V = 2 * π * (36/π)^(2/3) * (36/π)^(1/3)
V = 2 * π * (36/π)^(1)
V = 2 * π * 36/π
V = 72 m³
So, the volume of the cylinder is 72 m³.