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)

First, we need to find the radius and height of the cylinder. Since the volume of the cylinder is 36 cm^3, we can use the formula for the volume of a cylinder:

V = πr^2h

36 = πr^2h

Since the height of the cylinder is equal to the diameter of the sphere, we have h = 2r. Substituting this into the equation gives us:

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 find the volume of the sphere using the formula for the volume of a sphere:

V = (4/3)πr^3

Substituting r = (18/π)^(1/3) into the equation gives us:

V = (4/3)π((18/π)^(1/3))^3

V = (4/3)π(18/π)

V = 24 cm^3

Therefore, the volume of the sphere is 24 cm^3.

So, the correct answer is:
24 cm^3