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

To find the volume of the sphere given the volume of the cylinder, we'll first outline the relationships and formulas for each of the shapes.

1. **Volume of a Cylinder**: The volume \( V_c \) of a cylinder is given by the formula:
\[
V_c = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.

2. **Volume of a Sphere**: The volume \( V_s \) of a sphere is given by the formula:
\[
V_s = \frac{4}{3} \pi r^3
\]

Given that the volume of the cylinder \( V_c \) is \( 36 \, \text{cm}^3 \), we can write:
\[
\pi r^2 h = 36
\]

Next, we are told that the height of the cylinder is equal to the diameter of the sphere. Since the diameter of the sphere is \( 2r \), we have:
\[
h = 2r
\]

Substituting \( h = 2r \) into the volume formula for the cylinder, we get:
\[
\pi r^2 (2r) = 36
\]
This simplifies to:
\[
2\pi r^3 = 36
\]
Dividing both sides by \( 2\pi \):
\[
r^3 = \frac{36}{2\pi} = \frac{18}{\pi}
\]

Now, we can calculate \( r \):
\[
r = \left(\frac{18}{\pi}\right)^{1/3}
\]

Now we need to find the volume of the sphere using the radius \( r \). Plugging \( r^3 \) back into the sphere volume formula:
\[
V_s = \frac{4}{3} \pi r^3
\]
Substituting \( r^3 = \frac{18}{\pi} \):
\[
V_s = \frac{4}{3} \pi \left(\frac{18}{\pi}\right)
\]
Simplifying this:
\[
V_s = \frac{4}{3} \cdot 18 = \frac{72}{3} = 24
\]

Thus, the volume of the sphere is \( \boxed{24 \, \text{cm}^3} \).