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?(1 point)
Responses

24 cm3
24 cm cubed

12 cm3
12 cm cubed

18 cm3
18 cm cubed

54 cm3

Answer 1

To solve the problem, we need to use the formulas for the volumes of the cylinder, sphere, and relate their dimensions as given.

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

2. **Volume of a sphere (V_sphere)** is given by:
\[
V_{sphere} = \frac{4}{3} \pi r^3
\]

### Given:
- Volume of the cylinder, \( V_{cylinder} = 36 \, \text{cm}^3 \)
- The height of the cylinder is equal to the diameter of the sphere. Thus, \( h = 2r \).

### Setup:
We can substitute \( h \) into the volume of the cylinder:
\[
V_{cylinder} = \pi r^2 (2r) = 2\pi r^3
\]
Setting this equal to the given volume of the cylinder:
\[
2\pi r^3 = 36
\]

### Solve for \( r^3 \):
\[
\pi r^3 = 18 \quad \Rightarrow \quad r^3 = \frac{18}{\pi}
\]

### Now, let's find the volume of the sphere:
Substituting \( r^3 \) into the formula for the volume of the sphere:
\[
V_{sphere} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi \left(\frac{18}{\pi}\right) = \frac{4 \cdot 18}{3} = \frac{72}{3} = 24 \, \text{cm}^3
\]

### Conclusion:
The volume of the sphere is \( 24 \, \text{cm}^3 \).

### Responses:
24 cm³
24 cm cubed