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)
54 cm 3
24 cm 3
18 cm 3
12 cm 3
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