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?

answers:

24 cm^3

54 cm^3

18 cm^3

12 cm^3

Answer 1

To solve the problem, we first need to understand the relationships between the volumes of the cylinder, sphere, and cone.

1. **Volume of a Cylinder**: The formula for the volume \( V \) of a cylinder is given by:

\[
V = \pi r^2 h
\]

where \( r \) is the radius and \( h \) is the height.

2. **Volume of a Sphere**: The formula for the volume \( V \) of a sphere is given by:

\[
V = \frac{4}{3} \pi r^3
\]

3. **Given Data**: The volume of the cylinder is \( 36 \, \text{cm}^3 \) and the height of the cylinder \( h \) is equal to the sphere's diameter, which is \( 2r \).

Therefore, we can write the volume of the cylinder as:

\[
V = \pi r^2 (2r) = 2\pi r^3
\]

Setting this equal to the given volume of the cylinder:

\[
2\pi r^3 = 36
\]

Dividing both sides by \( 2\pi \):

\[
r^3 = \frac{36}{2\pi} = \frac{18}{\pi}
\]

Now, to find the volume of the sphere with this radius, we substitute \( r^3 \) back into the formula for the sphere's volume:

\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]

Thus,

\[
V_{\text{sphere}} = \frac{4}{3} \pi \left( \frac{18}{\pi} \right)
\]

Simplifying this gives:

\[
V_{\text{sphere}} = \frac{4 \times 18}{3} = \frac{72}{3} = 24 \, \text{cm}^3
\]

So, the volume of the sphere is **24 cm³**.