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

Answer 1

To find the volume of the tin man's head, we need to calculate the volumes of both the cylindrical part and the conical part and then add them together.

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

In this case:
- Radius \( r = 4 \) inches
- Height \( h = 12 \) inches

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

2. **Volume of the Cone**:
The formula for the volume of a cone is given by:
\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 h
\]

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

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

3. **Total Volume of the Tin Man's Head**:
Now we add the volumes of the cylinder and the cone:
\[
V_{\text{total}} = V_{\text{cylinder}} + V_{\text{cone}} = 192\pi + 32\pi = 224\pi
\]

Therefore, the volume of the tin man's head in terms of \(\pi\) is \( \boxed{224\pi} \).