Drainage tubing comes in large rolls. At your hardware store, you cut tubing to the lengths the customers want. You also provide customers with the volume of their tubing because they need to fill the tubing with gravel as they install it. The tubing's inside radius is 2 inches.

Which of the following is an expression for the volume of [tex]\(L\)[/tex] feet of drainage tubing, in cubic feet?

A. [tex]\(0.09 L\)[/tex]
B. [tex]\(3.14 L\)[/tex]
C. [tex]\(6.28 L\)[/tex]
D. [tex]\(12.56 L\)[/tex]
E. [tex]\(150.72 L\)[/tex]



Answer :

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]