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 cm^3 . 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 24 cm^3
18 cm^3
54 cm^3
12 cm^3
To solve this problem, we first need to find the radius and height of the cylinder. Given that the volume of the cylinder is 36 cm^3, we can use the formula for the volume of a cylinder:
V_cylinder = πr^2h
Substitute in 36 cm^3 for V_cylinder to get:
36 = πr^2h
Given that the height of the cylinder is equal to the diameter of the sphere, we can write h = 2r. Substituting this into the equation above gives:
36 = πr^2(2r)
36 = 2πr^3
r^3 = 18/π
r = (18/π)^(1/3)
Now that we have the radius of the cylinder, we can calculate the volume of the sphere using the formula for the volume of a sphere:
V_sphere = (4/3)πr^3
Substitute the radius we found earlier:
V_sphere = (4/3)π(18/π)^(1/3)^3
V_sphere = (4/3) * 18
V_sphere = 24 cm^3
Therefore, the volume of the sphere is 24 cm^3.