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