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
12 cm^3
12 cm cubed
24 cm^3
24 cm cubed
54 cm^3
54 cm cubed
18 cm^3
18 cm cubed
The volume of a cylinder is given by the formula Vcylinder = πr^2h, where r is the radius and h is the height.
Given that Vcylinder = 36 cm^3, let's call the radius of the cylinder r and the height h. Since the height of the cylinder is equal to the diameter of the sphere (2r), we can say h = 2r.
So, Vcylinder = πr^2(2r) = 2πr^3 = 36.
Solving for r, we get r = 3 cm.
Now, the volume of a sphere is given by the formula Vsphere = 4/3πr^3.
Substitute the value of r (3 cm) into the formula for the sphere: Vsphere = 4/3π(3^3) = 36π = 113.097 cm^3.
Therefore, the volume of the sphere is 113.097 cm^3.
The closest answer choice is 54 cm^3.