The volume of a sphere is 48 meters cubed. What is the volume of a cylinder if its radius Is the same as the spheres and its height Is equal to the spheres diameter?
The volume of a sphere is given by the formula V_sphere = (4/3)πr^3, where r is the radius of the sphere. We are given that the volume of the sphere is 48 meters cubed, so we can set up the equation as follows:
48 = (4/3)πr^3
To find the radius, we can rearrange the equation:
r^3 = (3/4)(48/π)
r^3 = 36/π
Taking the cube root of both sides:
r = (36/π)^(1/3)
Now, for the cylinder, the radius is the same as the sphere's radius, and the height is equal to the sphere's diameter. The diameter of the sphere is twice the radius, so the height of the cylinder is 2r. The volume of a cylinder is given by the formula V_cylinder = πr^2h, so substituting the values, we have:
V_cylinder = π[(36/π)^(1/3)]^2 * 2[(36/π)^(1/3)]
Simplifying,
V_cylinder = 2 * 36^(2/3)
Therefore, the volume of the cylinder is 2 * 36^(2/3) meters cubed.