Chapter 5.2, Problem 40PS

Solution

(a)

To graph: The function and state its domain and range.

Given information:

A business is studying the cost to remove a pollutant from the ground at its site. The function $y=\frac{15x}{1.1-x}$ models the estimated cost $y$ (in thousands of dollars) to remove $x$ percent (expressed as a decimal) of the pollutant.

Explanation:

Consider the given model $y=\frac{15x}{1.1-x}$.

It can also be written as $y=\frac{15x}{-x+1.1}$

The asymptotes are $x=1.1,y=-15$.

Graph the given model.

  

Interpretation:

The domain is a set of all real values except $1.1$ and the range is a set of real values except $-15$.

(b)

To calculate: The cost to remove $20\%,40\%\text{ and }80\%$ of pollutant and to state that whether the cost is also doubled if the pollutant removed is doubled.

The cost to remove $20\%,40\%\text{ and }80\%$ of pollutant is $3.33$ , $8.57$ and $40$ thousand dollars respectively and the cost is not doubled if the pollutant removed is doubled.

Given information:

A business is studying the cost to remove a pollutant from the ground at its site. The function $y=\frac{15x}{1.1-x}$ models the estimated cost $y$ (in thousands of dollars) to remove $x$ percent (expressed as a decimal) of the pollutant.

Calculation:

Consider the given model $y=\frac{15x}{1.1-x}$ , where $y$ represents the estimated cost and $x$ represents the percentage of pollutants in decimal.

To find the cost of removing $20\%$ of pollutant, substitute $x=0.2$.

  $\begin{array}{c}y=\frac{15\left(0.2\right)}{1.1-\left(0.2\right)}\\ =3.33\text{ thousand dollars}\end{array}$

To find the cost of removing $40\%$ of pollutant, substitute $x=0.4$.

  $\begin{array}{c}y=\frac{15\left(0.4\right)}{1.1-\left(0.4\right)}\\ =8.57\text{ thousand dollars}\end{array}$

To find the cost of removing $80\%$ of pollutant, substitute $x=0.8$.

  $\begin{array}{c}y=\frac{15\left(0.8\right)}{1.1-\left(0.8\right)}\\ =40\text{ thousand dollars}\end{array}$

Therefore, the cost to remove $20\%,40\%\text{ and }80\%$ of pollutant is $3.33$ , $8.57$ and $40$ thousand dollars respectively and the cost is not doubled if the pollutant removed is doubled.