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 cm3. 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?(1 point) Responses
54 cm^3
54 cm cubed
18 cm^3
18 cm cubed
12 cm^3
12 cm cubed
24 cm^3
The volume of a cylinder is given by the formula V_cylinder = πr^2h, where r is the radius and h is the height of the cylinder. Given that the volume of the cylinder is 36 cm^3 and the height is equal to the diameter of the sphere (2r), we can rewrite the formula as V_cylinder = πr^2(2r) = 2πr^3 = 36.
Solving for r, we get r = 3 cm.
The volume of a sphere is given by the formula V_sphere = 4/3πr^3. Substituting the radius for the sphere (r = 3 cm) into the formula, we get V_sphere = 4/3π(3)^3 = 36π cm^3.
Therefore, the volume of the sphere is 36π cm^3, which is approximately equal to 113.1 cm^3.