Question

A tin man has a head that is a cylinder with a cone on top. The height of the cylinder is 12 inches and the height of the cone is 6 inches. The radius of both the cylinder and the cone is 4 inches. What is the volume of the tin man's head in terms of pi?
Responses
A 192π
192π in 3
B 224π
224π in 3
C 384π
384π in 3
D 912π
912π in 3

Answer 1

To find the volume of the tin man's head, which consists of a cylinder and a cone, we need to calculate the volumes of both shapes separately and then add them together.

### Volume of the Cylinder
The formula for the volume \( V \) of a cylinder is given by:
\[
V = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.

For the cylinder:
- Radius \( r = 4 \) inches
- Height \( h = 12 \) inches

Substituting the values:
\[
V_{cylinder} = \pi (4)^2 (12) = \pi (16)(12) = 192\pi \, \text{cubic inches}
\]

### Volume of the Cone
The formula for the volume \( V \) of a cone is given by:
\[
V = \frac{1}{3} \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.

For the cone:
- Radius \( r = 4 \) inches
- Height \( h = 6 \) inches

Substituting the values:
\[
V_{cone} = \frac{1}{3} \pi (4)^2 (6) = \frac{1}{3} \pi (16)(6) = \frac{1}{3} \pi (96) = 32\pi \, \text{cubic inches}
\]

### Total Volume of the Tin Man's Head
Now, we sum the volumes of the cylinder and the cone:
\[
V_{total} = V_{cylinder} + V_{cone} = 192\pi + 32\pi = 224\pi \, \text{cubic inches}
\]

Thus, the volume of the tin man's head in terms of \( \pi \) is:
\[
\boxed{224\pi}
\]

So the correct answer is B: \( 224\pi \) cubic inches.