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?(1 point)
Responses

24 cm3
24 cm cubed

18 cm3
18 cm cubed

54 cm3
54 cm cubed

12 cm3

Answer 1

To solve the problem, we first need to establish the relationship between the volumes of a cylinder, sphere, and cone. The formulas for their volumes are:

1. **Volume of a cylinder**: \( V_{cylinder} = \pi r^2 h \)
2. **Volume of a sphere**: \( V_{sphere} = \frac{4}{3} \pi r^3 \)
3. **Volume of a cone**: \( V_{cone} = \frac{1}{3} \pi r^2 h \)

Given that the volume of the cylinder is \( 36 \, \text{cm}^3 \) and that the height of the cylinder is equal to the diameter of the sphere, we can set up the relationships as follows:

Let \( r \) be the radius of the cylinder (and sphere) and \( h \) be the height of the cylinder. Since the diameter of the sphere is twice the radius:

- The diameter of the sphere \( = 2r \)
- The height of the cylinder \( h = 2r \)

Substituting \( h = 2r \) into the volume formula for the cylinder gives:

\[
V_{cylinder} = \pi r^2 (2r) = 2\pi r^3
\]

We know the volume of the cylinder is \( 36 \, \text{cm}^3 \):

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

Now, we can solve for \( r^3 \):

\[
\pi r^3 = 18 \quad \Rightarrow \quad r^3 = \frac{18}{\pi}
\]

Now we can use this value of \( r \) to find the volume of the sphere:

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

Simplifying this gives:

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

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

The correct answer is \( \boxed{24 \, \text{cm}^3} \).