To find the total volume of the turret, we need to calculate the volumes of both the cylindrical part and the conical part and add them together.
Step 1: Calculate the 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 of the cylinder,
- \( h \) is the height of the cylinder.
The diameter of the turret is 15 feet, so the radius \( r \) is:
\[ r = \frac{15}{2} = 7.5 \text{ feet} \]
The height \( h \) of the cylinder is 40 feet. Now substituting the values into the volume formula:
\[ V_{\text{cylinder}} = \pi (7.5)^2 (40) \]
Calculating \( (7.5)^2 \):
\[ (7.5)^2 = 56.25 \]
Now substituting into the volume formula:
\[ V_{\text{cylinder}} = \pi (56.25)(40) \]
\[ V_{\text{cylinder}} = \pi (2250) \]
So,
\[ V_{\text{cylinder}} = 2250\pi \text{ cubic feet} \]
Step 2: Calculate the 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 of the cone (which is the same as the cylinder),
- \( h \) is the height of the cone.
The height of the cone is one-fourth the height of the cylinder:
\[ h_{\text{cone}} = \frac{1}{4} \times 40 = 10 \text{ feet} \]
Now using the radius \( r = 7.5 \) feet and \( h = 10 \) feet in the volume formula for the cone:
\[ V_{\text{cone}} = \frac{1}{3} \pi (7.5)^2 (10) \]
Using \( (7.5)^2 = 56.25 \):
\[ V_{\text{cone}} = \frac{1}{3} \pi (56.25)(10) \]
\[ V_{\text{cone}} = \frac{1}{3} \pi (562.5) = 187.5\pi \text{ cubic feet} \]
Step 3: Calculate the Total Volume
Now, add the volumes of the cylinder and the cone:
\[ V_{\text{total}} = V_{\text{cylinder}} + V_{\text{cone}} = 2250\pi + 187.5\pi = (2250 + 187.5)\pi = 2437.5\pi \text{ cubic feet} \]
Step 4: Calculate the Numerical Volume
Using the approximation \( \pi \approx 3.14 \):
\[ V_{\text{total}} \approx 2437.5 \times 3.14 \approx 7644.75 \]
Rounding to the nearest tenth, we find:
\[ \boxed{7644.8} \]
Thus, the total volume of the turret is approximately \( 7644.8 \) cubic feet.