Question

4. A contractor found that his labor cost for installing 80 feet of pipe was $20. He also found that his labor cost for installing 200 feet of pipe was $100. If the cost C in dollars is a linear function of the length L in feet, then what is the formula for this function? What would his labor cost be for installing 210 feet of pipe?

Answer 1

Since, the cost C in dollars is a linear function of the length L in feet, we can write:

$C=mL+b$

Where

m is the slope

b is the y-intercept of the line graphed.

The points are in the from (L, C) which is (length, cost). Given:

(80, 20) and (200, 100)

The slope (m) is:

$m=\frac{y_2-y_1}{x_2-x_1}$

Let the points be:

$\begin{gathered} (x_1,y_1)=(80,20) \\ (x_2,y_2)=(200,100) \end{gathered}$

So slope is:

$\begin{gathered} m=\frac{y_2-y_1}{x_2-x_1} \\ m=\frac{100-20_{}}{200-80_{}} \\ m=\frac{80}{120} \\ m=\frac{2}{3} \end{gathered}$

The equation becomes:

$C=\frac{2}{3}L+b$

Let's take the point (L, C) = (80, 20) and find out b:

$\begin{gathered} C=\frac{2}{3}L+b \\ 20=\frac{2}{3}(80)+b \\ 20=\frac{160}{3}+b \\ b=20-\frac{160}{3} \\ b=\frac{60-160}{3} \\ b=\frac{-100}{3} \end{gathered}$

The formula for the function is:

$C=\frac{2}{3}L-\frac{100}{3}$

The cost of installing 210 feet of pipe:

We plug in L = 210 into formula and find C:

$\begin{gathered} C=\frac{2}{3}(210)-\frac{100}{3} \\ C=140-\frac{100}{3} \\ C=\frac{420-100}{3} \\ C=\frac{320}{3} \\ C=\ 106.67 \end{gathered}$