Chapter 8.4, Problem 37PPS

a.

To determine

To write and graph the rational function representing Kristina’s average monthly cost as a function of number of months she uses the phone

Answer to Problem 37PPS

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:

Kristina bought a new cell phone cost $150 with internet access.

Her monthly usage charge is $30 plus $10 for the internet access

Asume that the number of months kristina uses her phone as $\text{'}x\text{'}$

Thus, the total cost on usage of cell phone is $150+\left(30+10\right)x$

This means that the average cost for $x$ usage of cell phone is,

  $\begin{array}{l}\frac{150+\left(30+10\right)x}{x}\\ \frac{150+40x}{x}\end{array}$

Therefore, the rational function that represents Kristina’s average monthly cost as a function of the number of months she uses the phone is,

  $f\left(x\right)=\frac{150+40x}{x}$

The graph of the function can be represented as,

  

b.

To determine

To find out the asymptotes of the graph

Answer to Problem 37PPS

The vertical asymptote is $x=0$

Explanation of Solution

Given information:

Kristina bought a new cell phone cost $150 with internet access.

Her monthly usage charge is $30 plus $10 for the internet access

The rational function that represents Kristina’s average monthly cost as a function of the number of months she uses the phone is,

  $f\left(x\right)=\frac{150+40x}{x}$ Here, $\text{'}x\text{'}$ is the number of months Kristina uses her phone

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 37PPS

The first quadrant in the graph is only relevant because $\text{'}x\text{'}$ represents a number of months Kristina uses the phone, thus $\text{'}x\text{'}$ is positive number and its value cannot be negative

Explanation of Solution

Given information:

Kristina bought a new cell phone cost $150 with internet access.

Her monthly usage charge is $30 plus $10 for the internet access

At first note that the number of months Kristina uses the phone is $\text{'}x\text{'}$ . In part $\left(a\right)$ it is determined that the rational function modelling the average monthly cost is,

  $f\left(x\right)=\frac{150+40x}{x}$

Therefore, it can be said that only first quadrant is relevant because $\text{'}x\text{'}$ represents a number of months Kristina uses the phone, thus $\text{'}x\text{'}$ is positive number and its value cannot be negative

d.

To determine

To find out that after how many months the average monthly charge will be $45

Answer to Problem 37PPS

The average monthly cost will be $45 after 30 months

Explanation of Solution

Given information:

Kristina bought a new cell phone cost $150 with internet access.

Her monthly usage charge is $30 plus $10 for the internet access

The average monthly cost is given by the function $f\left(x\right)=\frac{150+40x}{x}$

To find the number of months it will take for the monthly average cost to be $45, set $f\left(x\right)$ to 45

Now the solution for $x$ can be obtained as,

  $\begin{array}{l}f\left(x\right)=\frac{150+40x}{x}\\ 45=\frac{150+40x}{x}\\ x\left(45\right)=150+40x\\ 45x-40x=150\\ 5x=150\\ x=\frac{150}{5}\\ x=30\end{array}$

Therefore, the average monthly cost will be $45 after 30 months