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
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?(1 point)
Responses

54 cm3
54
cm
3
54 cm cubed

18 cm3
18
cm
3
18 cm cubed

12 cm3
12
cm
3
12 cm cubed

24 cm3

Answer 1

To solve this problem, we will use the relationships between the volumes of cones, cylinders, and spheres.

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

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

In this case:
- The volume of the cylinder is given: \(V_{cylinder} = 36 \, \text{cm}^3\).
- We need to relate the dimensions: the height of the cylinder \(h\) is equal to the sphere's diameter. Since the diameter of the sphere is \(2r\), we have \(h = 2r\).

3. Substitute \(h = 2r\) into the volume of the cylinder formula:
\[
V_{cylinder} = \pi r^2 (2r) = 2 \pi r^3
\]
Set this equal to the volume of the cylinder:
\[
2 \pi r^3 = 36
\]
Solve for \(r^3\):
\[
r^3 = \frac{36}{2 \pi} = \frac{18}{\pi}
\]

4. Now, we can find the volume of the sphere using the derived value of \(r\):
\[
V_{sphere} = \frac{4}{3} \pi r^3
\]
Substitute \(r^3\) into the volume of the sphere formula:
\[
V_{sphere} = \frac{4}{3} \pi \left(\frac{18}{\pi}\right) = \frac{4 \cdot 18}{3} = \frac{72}{3} = 24 \, \text{cm}^3
\]

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