To find the volume of a right circular cone, we need to use the formula for the volume of a cone:
\(\[ V = \frac{1}{3} \pi r^2 h \]\)
where \(\( V \)\) is the volume, \(\( r \)\) is the radius of the base, and \(\( h \)\) is the height.
Here are the steps to solve the problem:
1. Identify the given dimensions:
- Height (\(\( h \)\)) = 13.1 inches
- Diameter (\(\( d \)\)) = 5.7 inches
2. Calculate the radius (\(\( r \)\)):
The radius is half of the diameter. Therefore,
\(\[
r = \frac{d}{2} = \frac{5.7}{2} = 2.85 \text{ inches}
\]\)
3. Plug the values into the volume formula:
Now that we know the radius and the height, we can substitute these values into the volume formula.
\(\[
V = \frac{1}{3} \pi (2.85)^2 (13.1)
\]\)
4. Calculate the volume:
First, calculate the square of the radius:
\(\[
(2.85)^2 = 8.1225
\]\)
Then multiply this by the height:
\(\[
8.1225 \times 13.1 = 106.40475
\]\)
Finally, multiply by \(\(\pi\)\) and \(\(\frac{1}{3}\)\):
\(\[
V = \frac{1}{3} \pi \times 106.40475 \approx 111.42679363568617 \text{ cubic inches}
\]\)
5. Round the volume to the nearest tenth:
The volume rounded to the nearest tenth is:
\(\[
V \approx 111.4 \text{ cubic inches}
\]\)
Therefore, the volume of the right circular cone is approximately \(\( 111.4 \)\) cubic inches.
