Certainly! Let's tackle the problem of finding the volume of a tree stump modeled as a right cylinder given its height in feet and its circumference in inches. To do this, we need to follow these steps:
1. Convert the height from feet to inches.
2. Find the radius of the cylinder from its circumference.
3. Use the formula for the volume of a cylinder to find the volume.
### Step-by-Step Solution
1. Convert the height from feet to inches:
- The height provided is \(\( 1.6 \)\) feet.
- We know that \(\( 1 \)\) foot is equal to \(\( 12 \)\) inches.
- Therefore, the height in inches is:
\(\[
\text{Height in inches} = 1.6 \times 12 = 19.2 \text{ inches}
\]\)
2. Find the radius from the circumference:
- The circumference provided is \(\( 61 \)\) inches.
- The formula for the circumference \(\( C \)\) of a circle is:
\(\[
C = 2 \pi r
\]\)
where \(\( r \)\) is the radius.
- We need to solve for \(\( r \)\):
\(\[
r = \frac{C}{2 \pi} = \frac{61}{2 \pi}
\]\)
- Plug in the value of \(\( \pi \)\) (approximately \(\( 3.1416 \)\)):
\(\[
r = \frac{61}{2 \times 3.1416} \approx \frac{61}{6.2832} \approx 9.7 \text{ inches}
\]\)
3. Calculate the volume of the cylinder:
- The formula for the volume \(\( V \)\) of a cylinder is:
\(\[
V = \pi r^2 h
\]\)
where \(\( r \)\) is the radius and \(\( h \)\) is the height.
- Substituting the radius \(\( r \approx 9.7 \)\) inches and height \(\( h = 19.2 \)\) inches:
\(\[
V = \pi (9.7)^2 \times 19.2
\]\)
- Calculate \(\( 9.7^2 \)\):
\(\[
9.7^2 = 94.09
\]\)
- Now calculate the volume:
\(\[
V = \pi \times 94.09 \times 19.2 \approx 3.1416 \times 94.09 \times 19.2 \approx 3.1416 \times 1805.728
\]\)
\(\[
V \approx 5672.3 \text{ cubic inches}
\]\)
### Final Answer
The volume of the tree stump is approximately \(\( 5672.3 \)\) cubic inches, rounded to the nearest tenth.
