Chapter 8.4, Problem 36PPS

a.

To determine

To write and graph the rational function representing Liam’s average cost per customer as a function of number of parking lots

Answer to Problem 36PPS

The rational function that represents Liam’s average cost per customer as a function of number of parking lots is $f\left(x\right)=\frac{50x+4500}{x}$ and graph for this function is represented

Explanation of Solution

Given information:

Liam purchased a snow plow for $\$4500$ and he incurs a cost of $50 for gas and maintenance when he plows the parking lots each time.

The snow plow cost $4500 and he incurs a cost of $50 for gas and maintenance when he mows a lawn.

Thus, the total cost on Liam’s part is $50x+4500$

This means that the average cost for $x$ customers is,

  $\frac{50x+4500}{x}$

Therefore, the function that represents his average cost per customer as a function of the number of parking lots mowed is,

  $f\left(x\right)=\frac{50x+4500}{x}$

The graph of the function can be represented as,

  

b.

To determine

To find out the asymptotes of the graph

Answer to Problem 36PPS

The vertical asymptote is $x=0$

Explanation of Solution

Given information:

Liam purchased a snow plow for $\$4500$ and he incurs a cost of $50 for gas and maintenance when he plows the parking lots each time.

Rational function of average cost per parking lot is $f\left(x\right)=\frac{50x+4500}{x}$

To find the vertical asymptotes, set the denominator equal to zero

So, put $x=0$

Therefore, the vertical asymptote is $x=0$

c.

To determine

To explain why the first quadrant in the graph is the only relevant quadrant

Answer to Problem 36PPS

The first quadrant in the graph is only relevant because $x$ represents the number of parking lawns mowed and its value cannot be negative

Explanation of Solution

Given information:

Liam purchased a snow plow for $\$4500$ and he incurs a cost of $50 for gas and maintenance when he plows the parking lots each time.

The number of parking lots mowed is $x$ .

Thus, its value cannot be negative.

Hence, the only part of the graph that is relevant is the one in the first quadrant.

d.

To determine

To find out the total number of parking lot Liam need to plow for his average cost per parking lot is less than $80

Answer to Problem 36PPS

Liam need to plow less than 150 times for his average cost per parking lot to be less than $80

Explanation of Solution

Given information:

Liam purchased a snow plow for $\$4500$ and he incurs a cost of $50 for gas and maintenance when he plows the parking lots each time.

Rational function of average cost per parking lot is $f\left(x\right)=\frac{50x+4500}{x}$

For the cost to be less than $80 the function must be written as,

  $\begin{array}{l}f\left(x\right)<80\\ \frac{4500+50x}{x}<80\end{array}$

The obtained function must be solved as,

  $\begin{array}{l}80-\frac{4500+50x}{x}>0\\ \frac{80x-4500-50x}{x}>0\\ \frac{30x-4500}{x}>0\\ \frac{30\left(x-150\right)}{x}>0\\ \left(x-150\right)x>0\\ \Rightarrow x\in \left(-\infty ,0\right)\cup \left(150,\infty \right)\end{array}$

As $x>0$ , it means the solution is $x>150$