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 [tex]\( 1.6 \)[/tex] feet.
- We know that [tex]\( 1 \)[/tex] foot is equal to [tex]\( 12 \)[/tex] inches.
- Therefore, the height in inches is:
[tex]\[
\text{Height in inches} = 1.6 \times 12 = 19.2 \text{ inches}
\][/tex]
2. Find the radius from the circumference:
- The circumference provided is [tex]\( 61 \)[/tex] inches.
- The formula for the circumference [tex]\( C \)[/tex] of a circle is:
[tex]\[
C = 2 \pi r
\][/tex]
where [tex]\( r \)[/tex] is the radius.
- We need to solve for [tex]\( r \)[/tex]:
[tex]\[
r = \frac{C}{2 \pi} = \frac{61}{2 \pi}
\][/tex]
- Plug in the value of [tex]\( \pi \)[/tex] (approximately [tex]\( 3.1416 \)[/tex]):
[tex]\[
r = \frac{61}{2 \times 3.1416} \approx \frac{61}{6.2832} \approx 9.7 \text{ inches}
\][/tex]
3. Calculate the volume of the cylinder:
- The formula for the volume [tex]\( V \)[/tex] of a cylinder is:
[tex]\[
V = \pi r^2 h
\][/tex]
where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height.
- Substituting the radius [tex]\( r \approx 9.7 \)[/tex] inches and height [tex]\( h = 19.2 \)[/tex] inches:
[tex]\[
V = \pi (9.7)^2 \times 19.2
\][/tex]
- Calculate [tex]\( 9.7^2 \)[/tex]:
[tex]\[
9.7^2 = 94.09
\][/tex]
- Now calculate the volume:
[tex]\[
V = \pi \times 94.09 \times 19.2 \approx 3.1416 \times 94.09 \times 19.2 \approx 3.1416 \times 1805.728
\][/tex]
[tex]\[
V \approx 5672.3 \text{ cubic inches}
\][/tex]
### Final Answer
The volume of the tree stump is approximately [tex]\( 5672.3 \)[/tex] cubic inches, rounded to the nearest tenth.
