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. (1 point)

Answer 1

To find the volume of the turret, we need to calculate the volumes of both the cylindrical part and the conical part, and then 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 and \( h \) is the height of the cylinder.

Given that the diameter of the turret is 15 feet, the radius \( r \) is:

\[
r = \frac{15}{2} = 7.5 \text{ feet}
\]

The height \( h \) of the cylinder is given as 40 feet. Thus, the volume of the cylinder is:

\[
V_{\text{cylinder}} = \pi (7.5)^2 (40)
\]

Calculating \( (7.5)^2 \):

\[
(7.5)^2 = 56.25
\]

Now plugging this into the volume formula:

\[
V_{\text{cylinder}} = \pi (56.25) (40) = 2250\pi \text{ cubic feet}
\]

### Step 2: Calculate the volume of the cone.

The height of the cone is one-fourth the height of the cylinder:

\[
h_{\text{cone}} = \frac{1}{4} \times 40 = 10 \text{ feet}
\]

The formula for the volume \( V \) of a cone is given by:

\[
V = \frac{1}{3} \pi r^2 h
\]

Using the same radius \( r = 7.5 \text{ feet} \) and height \( h_{\text{cone}} = 10 \text{ feet} \):

\[
V_{\text{cone}} = \frac{1}{3} \pi (7.5)^2 (10)
\]

Using \( (7.5)^2 = 56.25 \):

\[
V_{\text{cone}} = \frac{1}{3} \pi (56.25) (10) = \frac{1}{3} \pi (562.5) = 187.5\pi \text{ cubic feet}
\]

### Step 3: Calculate the total volume of the turret.

Now, we sum the volumes of the cylinder and cone:

\[
V_{\text{turret}} = V_{\text{cylinder}} + V_{\text{cone}} = 2250\pi + 187.5\pi = (2250 + 187.5)\pi = 2437.5\pi \text{ cubic feet}
\]

### Step 4: Evaluate the total volume.

Using \( \pi \approx 3.14159265 \):

\[
V_{\text{turret}} \approx 2437.5 \times 3.14159265 \approx 7665.1651
\]

### Step 5: Round to the nearest tenth.

\[
V_{\text{turret}} \approx 7665.2 \text{ cubic feet}
\]

Therefore, the volume of the turret is approximately:

\[
\boxed{7665.2}
\] cubic feet.