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. 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?
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.
Given that the volume of the cylinder is 36 cm, we can find the radius using the formula Vcylinder = πr^2h.
36 = πr^2h
We also know that the height of the cylinder is equal to the diameter of the sphere, so h = 2r.
Substituting h = 2r into the equation, we get:
36 = πr^2(2r)
36 = 2πr^3
Simplifying the equation, we have:
18 = πr^3
Now, we can divide both sides by π to isolate r^3:
18/π = r^3
Taking the cube root of both sides, we get:
r = ∛(18/π)
Now that we know the radius, we can calculate the volume of the sphere using the formula Vsphere = (4/3)πr^3:
Vsphere = (4/3)π(∛(18/π))^3
Vsphere = (4/3)π(∛(18))^3
Vsphere = (4/3)π(2∛2)^3
Vsphere = (4/3)π(8∛2)
Vsphere = (32/3)π∛2
Therefore, the volume of the sphere is (32/3)π∛2 cubic units.