Let's approach this problem step-by-step to find the correct expression for the volume of [tex]\( L \)[/tex] feet of drainage tubing with a radius of 2 inches.
1. Convert the radius from inches to feet:
- Given the radius [tex]\( r \)[/tex] is 2 inches.
- There are 12 inches in one foot.
- Therefore, the radius in feet is [tex]\(\frac{2}{12}\)[/tex] feet or [tex]\(\frac{1}{6}\)[/tex] feet.
2. Volume formula for a cylinder:
- The volume [tex]\( V \)[/tex] of a cylinder is given by [tex]\( V = \pi r^2 h \)[/tex], where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height or length of the cylinder.
3. Substitute the values:
- Here, the length [tex]\( h \)[/tex] of the tubing is [tex]\( L \)[/tex] feet.
- The radius [tex]\( r = \frac{1}{6} \)[/tex] feet.
So, the volume per foot [tex]\( V \)[/tex] can be calculated as:
[tex]\[
V = \pi \left(\frac{1}{6}\right)^2 \times L
\][/tex]
4. Simplify the expression:
- [tex]\[
\left(\frac{1}{6}\right)^2 = \frac{1}{36}
\][/tex]
- So,
[tex]\[
V = \pi \cdot \frac{1}{36} \cdot L
\][/tex]
- [tex]\[
V = \frac{\pi}{36} \cdot L
\][/tex]
5. Approximate the value of [tex]\(\frac{\pi}{36}\)[/tex]:
- [tex]\(\pi \approx 3.14159\)[/tex]
- [tex]\[
\frac{\pi}{36} \approx \frac{3.14159}{36} \approx 0.08726646259971647
\][/tex]
Therefore, the volume [tex]\( V \)[/tex] of [tex]\( L \)[/tex] feet of drainage tubing, in cubic feet, is approximately:
[tex]\[
V \approx 0.08726646259971647 \times L
\][/tex]
6. Identify the closest expression:
- Among the provided options, 0.08726646259971647 is very close to Option A: [tex]\( 0.09 \times L \)[/tex].
Thus, the expression that represents the volume of [tex]\( L \)[/tex] feet of drainage tubing, in cubic feet, is:
[tex]\[
\boxed{0.09 L}
\][/tex]
