To solve the problem of finding the approximate volume of the tree trunk, we need to follow these steps:
1. Calculate the Diameter of the Trunk:
- We are given that the circumference [tex]\( C \)[/tex] of the trunk is 4.5 feet.
- The formula for the circumference of a circle is [tex]\( C = \pi \times d \)[/tex], where [tex]\( d \)[/tex] is the diameter.
- Rearranging the formula to solve for [tex]\( d \)[/tex], we get [tex]\( d = \frac{C}{\pi} \)[/tex].
Substituting the given circumference into the formula:
[tex]\[
d = \frac{4.5}{\pi}
\][/tex]
[tex]\[
d \approx 1.432 \text{ feet}
\][/tex]
2. Calculate the Radius of the Trunk:
- The radius [tex]\( r \)[/tex] is half of the diameter.
[tex]\[
r = \frac{d}{2}
\][/tex]
[tex]\[
r \approx \frac{1.432}{2}
\][/tex]
[tex]\[
r \approx 0.716 \text{ feet}
\][/tex]
3. Calculate the Volume of the Trunk:
- The trunk is approximated as a cylinder. The formula for the volume [tex]\( V \)[/tex] of a cylinder is [tex]\( V = \pi \times r^2 \times h \)[/tex], where [tex]\( h \)[/tex] is the height.
- We are given the height [tex]\( h \)[/tex] is 13 feet.
Substituting the radius and height into the volume formula:
[tex]\[
V = \pi \times (0.716)^2 \times 13
\][/tex]
[tex]\[
V \approx 20.948 \text{ cubic feet}
\][/tex]
4. Match the Calculated Volume to the Closest Option:
- We compare the calculated volume [tex]\( 20.948 \)[/tex] cubic feet against the given options:
- A. [tex]\( 17.16 \, \text{ft}^3 \)[/tex]
- B. [tex]\( 20.95 \, \text{ft}^3 \)[/tex]
- C. [tex]\( 27.14 \, \text{ft}^3 \)[/tex]
- D. [tex]\( 33.09 \, \text{ft}^3 \)[/tex]
- The closest option to our calculated volume [tex]\( 20.948 \, \text{cubic feet} \)[/tex] is B. [tex]\( 20.95 \, \text{cubic feet} \)[/tex].
Therefore, the approximate volume of the tree trunk is:
B. [tex]\( 20.95 \, \text{ft}^3 \)[/tex].
