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?(1 point)
Responses

24 cm3
24 cm cubed

12 cm3
12 cm cubed

18 cm3
18 cm cubed

54 cm^3

Answer 1

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

- The volume \(V_c\) of a cylinder is given by:
\[
V_c = \pi r^2 h
\]

- The volume \(V_s\) of a sphere is given by:
\[
V_s = \frac{4}{3} \pi r^3
\]

- The volume \(V_{cone}\) of a cone is given by:
\[
V_{cone} = \frac{1}{3} \pi r^2 h
\]

Now, according to the problem:

1. The volume of the cylinder is \(36 \, \text{cm}^3\).
2. The sphere has the same radius \(r\) as the cylinder.
3. The height of the cylinder \(h\) is equal to the diameter of the sphere, which can be expressed as \(2r\).

Since we know the volume of the cylinder:
\[
V_c = \pi r^2 h = 36
\]

Substituting \(h\) with \(2r\):
\[
V_c = \pi r^2 (2r) = 2\pi r^3 = 36
\]

To find \(r^3\):
\[
2\pi r^3 = 36
\]
\[
r^3 = \frac{36}{2\pi} = \frac{18}{\pi}
\]

Now we will use this value of \(r^3\) to find the volume of the sphere:
\[
V_s = \frac{4}{3} \pi r^3
\]
\[
V_s = \frac{4}{3} \pi \left(\frac{18}{\pi}\right)
\]

Now, simplify the expression for \(V_s\):
\[
V_s = \frac{4 \cdot 18}{3} = \frac{72}{3} = 24 \, \text{cm}^3
\]

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