Use the relationship among the formulas for the volumes of cones, cylinders, and spheres to solve the following problem. The volume of a cylinder is 36 m³. What is the volume of a sphere if its radius is the same as the cylinder’s and the height of the cylinder is equal to the sphere’s diameter?
First, we need to find the radius and height of the cylinder. We know that the volume of a cylinder is given by the formula V_cylinder = πr^2h, where r is the radius and h is the height. We are given that V_cylinder = 36 m³.
Using the formula for the volume of a cylinder, we can rearrange it to solve for h:
V_cylinder = πr^2h
36 = πr^2h
h = 36 / (πr^2)
We are also given that the height of the cylinder is equal to the diameter of the sphere, which means h = 2r.
Substituting h = 2r into the equation above, we get:
2r = 36 / (πr^2)
2r^3 = 36 / π
r^3 = 18 / π
r = (18 / π)^(1/3)
Now that we have found the radius of the cylinder, we can use the formula for the volume of a sphere V_sphere = 4/3 * πr^3 to find the volume of the sphere:
V_sphere = 4/3 * π * [(18 / π)^(1/3)]^3
V_sphere = 4/3 * π * 18
V_sphere = 24π
Therefore, the volume of the sphere is 24π m³.