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. 18cm^3
2. 24cm^3
3. 54cm^3
4. 12cm^3

Answer 1

To solve this problem, let's start by recalling the formulas for the volumes of a cylinder, sphere, and cone:

1. Volume of a cylinder (\( V_c \)):
\[
V_c = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.

2. Volume of a sphere (\( V_s \)):
\[
V_s = \frac{4}{3} \pi r^3
\]
where \( r \) is the radius.

3. Volume of a cone (\( V_{co} \)):
\[
V_{co} = \frac{1}{3} \pi r^2 h
\]

Given that the volume of the cylinder is 36 cm³ and the height of the cylinder (\( h \)) is equal to the diameter of the sphere, we can express these relationships stepwise.

1. **Volume of the Cylinder:**
\[
V_c = \pi r^2 h = 36 \, \text{cm}^3
\]

2. **Relating Cylinder Height and Sphere Radius:**
The height of the cylinder is equal to the diameter of the sphere:
\[
h = 2r
\]
Now, we can substitute \( h \) back into the cylinder volume formula:
\[
\pi r^2 (2r) = 36
\]
Simplifying this:
\[
2\pi r^3 = 36
\]
\[
\pi r^3 = 18
\]
\[
r^3 = \frac{18}{\pi}
\]

3. **Finding the Volume of the Sphere:**
Now we will now substitute \( r \) into the sphere volume formula:
\[
V_s = \frac{4}{3} \pi r^3
\]
Replacing \( r^3 \) with \( \frac{18}{\pi} \):
\[
V_s = \frac{4}{3} \pi \left(\frac{18}{\pi}\right)
\]
Simplifying this:
\[
V_s = \frac{4 \times 18}{3} = \frac{72}{3} = 24 \, \text{cm}^3
\]

Thus, the volume of the sphere is **24 cm³**, which corresponds to option **2. 24 cm³**.