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.