Question

A castle has a turret in the shape of a cylinder with a cone on top. The diameter of the turret is 15 feet, and the height of the cylinder is 40 feet. What is the volume if the height of the cone is exactly one-fourth the height of the cylinder? Enter your answer to the nearest tenth.

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 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.