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)Responses18 cm318 cm cubed12 cm312 cm cubed54 cm354 cm cubed24 cm3
First, let's recall the formulas for the volumes of a cylinder, cone, and sphere:
Cylinder: V = πr^2h
Sphere: V = (4/3)πr^3
Given that the volume of the cylinder is 36 cm^3, we can set up an equation to find the radius of the cylinder:
36 = πr^2h
36 = πr^2(2r)
36 = 2πr^3
r^3 = 18
r = ∛18 ≈ 2.62 cm
Since the height of the cylinder is equal to the diameter of the sphere, the diameter of the sphere is 2r = 5.24 cm, and the radius is half of that: 2.62 cm.
Now, we can find the volume of the sphere using the formula:
V (sphere) = (4/3)π(2.62)^3
V (sphere) ≈ 57.49 cm^3
Therefore, the volume of the sphere is approximately 57.49 cm^3.