Question

A castle has a turret in the shape of a cylinder with a cone on top. The diameter of the turret is 12 feet, and the height of the cylinder is 44 feet. %0D%0A%0D%0A%0D%0A%0D%0AWhat is the total volume if the height of the cone is exactly one-fourth the height of the cylinder?

Answer 1

To find the total volume of the turret, we need to calculate the volumes of both the cylindrical part and the conical part separately, then add them together.

1. **Dimensions of the turret:**
- Diameter of the turret: 12 feet
- Radius \( r \) of the turret:
\[
r = \frac{12}{2} = 6 \text{ feet}
\]
- Height of the cylinder \( h_c \): 44 feet
- Height of the cone \( h_{cone} \):
\[
h_{cone} = \frac{1}{4} \times h_c = \frac{1}{4} \times 44 = 11 \text{ feet}
\]

2. **Volume of the cylinder \( V_c \):**
The formula for the volume of a cylinder is:
\[
V_c = \pi r^2 h_c
\]
Substituting the values:
\[
V_c = \pi (6)^2 (44) = \pi (36)(44) = 1584\pi \text{ cubic feet}
\]

3. **Volume of the cone \( V_{cone} \):**
The formula for the volume of a cone is:
\[
V_{cone} = \frac{1}{3} \pi r^2 h_{cone}
\]
Substituting the values:
\[
V_{cone} = \frac{1}{3} \pi (6)^2 (11) = \frac{1}{3} \pi (36)(11) = \frac{396}{3}\pi = 132\pi \text{ cubic feet}
\]

4. **Total volume of the turret \( V_{total} \):**
Now we add the volumes of the cylinder and the cone together:
\[
V_{total} = V_c + V_{cone} = 1584\pi + 132\pi = (1584 + 132)\pi = 1716\pi \text{ cubic feet}
\]

5. **Final result:**
The total volume of the turret is:
\[
V_{total} \approx 1716 \times 3.14159 \approx 5397.96 \text{ cubic feet}
\]
Therefore, the exact volume is:
\[
V_{total} = 1716\pi \text{ cubic feet}
\]

Thus, the total volume of the turret is \(\mathbf{1716\pi \text{ cubic feet}}\) or approximately \(\mathbf{5397.96 \text{ cubic feet}}\).