Question

Question 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?

Answer 1

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

We also know that the height of the cylinder is equal to the sphere's diameter, which means h = 2r.

So the volume of the cylinder can be rewritten as Vcylinder = πr^2(2r) = 2πr^3.

Since the volume of the cylinder is 36 cm^3, we can set up the equation 2πr^3 = 36.

Now we can solve for r:

2πr^3 = 36
r^3 = 18/π
r ≈ ∛(18/π)

Once we have found the value of r, we can use it to find the volume of the sphere using the formula Vsphere = (4/3)πr^3:

Volume of sphere = Vsphere = (4/3)πr^3
Volume of sphere ≈ (4/3)π(∛(18/π))^3

Now we can plug in the value of π and evaluate:

Volume of sphere ≈ (4/3)(3.14159)(∛(18/3.14159))^3
Volume of sphere ≈ (4/3)(3.14159)(∛(5.73102))^3

Calculating this expression will give us the volume of the sphere, which measures approximately 57.28 cm^3.