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 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?

Answer 1

To solve the problem, we will start by recalling the formulas for the volumes of a cylinder, a sphere, and a cone:

1. **Volume of a cylinder**:
\[
V_{cylinder} = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.

2. **Volume of a sphere**:
\[
V_{sphere} = \frac{4}{3} \pi r^3
\]

3. **Volume of a cone**:
\[
V_{cone} = \frac{1}{3} \pi r^2 h
\]

From the problem statement, we know that the volume of the cylinder is \( 36 \, \text{cm}^3 \). We will set up the equations using the given information:

Let the radius of the cylinder (and sphere) be \( r \) and the height of the cylinder be \( h \). According to the problem, the height \( h \) of the cylinder is equal to the sphere's diameter. Since the diameter of a sphere is \( 2r \), we have:
\[
h = 2r
\]

Now we can substitute \( h \) in the volume formula of the cylinder:
\[
V_{cylinder} = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3
\]

Set this equal to the given volume of the cylinder:
\[
2\pi r^3 = 36
\]

Now, we can solve for \( r^3 \):
\[
r^3 = \frac{36}{2\pi} = \frac{18}{\pi}
\]

Next, we will substitute \( r^3 \) into the volume formula for the sphere:
\[
V_{sphere} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi \left(\frac{18}{\pi}\right)
\]

Simplifying the sphere's volume:
\[
V_{sphere} = \frac{4}{3} \cdot 18 = \frac{72}{3} = 24 \, \text{cm}^3
\]

Thus, the volume of the sphere is:
\[
\boxed{24 \, \text{cm}^3}
\]