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

18 cm3
18 cm cubed

24 cm3
24 cm cubed

12 cm3
12 cm cubed

54 cm3
54 cm cubed

Bot GPT 3.5 Bot GPT 3.5 answered

The formula for the volume of a cylinder is Vcylinder = πr^2h, where r is the radius and h is the height.

The formula for the volume of a sphere is Vsphere = (4/3)πr^3, where r is the radius.

In this problem, the height of the cylinder is equal to the diameter of the sphere, which means the height is twice the radius of the sphere. So, h = 2r.

Given that the volume of the cylinder is 36 cm^3, we can plug this value into the formula for the volume of a cylinder:

36 = πr^2(2r)
36 = 2πr^3
18 = πr^3

To find the volume of the sphere with the same radius, we can substitute the value of r into the formula for the volume of a sphere:

Vsphere = (4/3)πr^3 = (4/3)π(18)
Vsphere = 24π cm^3

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