Chapter 11.2, Problem 25E

(a)

To determine

To find:

The exponential function to model the percent of impurity.

Answer to Problem 25E

  $y={\left(0.85\right)}^3$

Explanation of Solution

Given:

Theimpurity is removed

  $15\%$

Concept used:

Formula

The kerosene left after $x\text{feet}$

  $y={\left(0.85\right)}^x$

Calculation:

The kerosene left after $1\text{feet}$

  $\begin{array}{l}=1-0.15\\ =0.85\end{array}$

The kerosene left after $2\text{feet}$

$0.85\times$ Kerosene left after $1\text{feet}$

  $\begin{array}{l}=0.85\times 0.85\\ ={\left(0.85\right)}^2\end{array}$

The kerosene left after $3\text{feet}$

$0.85\times$ Kerosene left after $2\text{feet}$

  $\begin{array}{l}=0.85\times 0.85\times 0.85\\ ={\left(0.85\right)}^3\end{array}$

(b)

To determine

To draw:

The graph of the second coordinate plane.

Explanation of Solution

Given:

The impurity is removed

  $15\%$

Concept used:

Replace the inequality sign and sketch the graph of the resulting equation (use a dashed line for < or > and a solid line for $\le or\ge$

Test one point in each of the region formed by the graph

If the point satisfies the inequality then shade the entire region to denote that every point in the region satisfies the inequality

Calculation:

Test one point in each of the region formed by the graph

If the point satisfies the inequality then shade the entire region to denote that every point in the region satisfies the inequality

Draw the table

$x$	$0$	$1$	$2$	$3$	$4$	$5$
$y$	$1$	$0.85$	$0.72$	$0.61$	$0.52$	$0.44$

Draw the graph

  $y={\left(0.85\right)}^x$

(c)

To determine

To find:

Theimpurity remains after the kerosene travels

Answer to Problem 25E

  $y\approx 14\%$

Explanation of Solution

Given:

The impurity is

  $12\text{feet}$

Concept used:

Formula

The kerosene left after $x\text{feet}$

  $y={\left(0.85\right)}^x$

Calculation:

The kerosene left after $1\text{feet}$

  $\begin{array}{l}=1-0.15\\ =0.85\end{array}$

The kerosene left after $2\text{feet}$

$0.85\times$ Kerosene left after $1\text{feet}$

  $\begin{array}{l}=0.85\times 0.85\\ ={\left(0.85\right)}^2\end{array}$

The kerosene left after $3\text{feet}$

$0.85\times$ Kerosene left after $2\text{feet}$

  $\begin{array}{l}=0.85\times 0.85\times 0.85\\ ={\left(0.85\right)}^3\end{array}$

The kerosene left after $12\text{feet}$

  $\begin{array}{l}y={\left(0.85\right)}^x\\ y={\left(0.85\right)}^{12}\\ y=0.142\\ y\approx 14\%\end{array}$

(d)

To determine

To explain:

The exponential function to model of the impurity ever completely removed.

Answer to Problem 25E

No solution

Explanation of Solution

Given:

The impurity is removed

  $15\%$

Concept used:

Formula

The kerosene left after $x\text{feet}$

  $y={\left(0.85\right)}^x$

Calculation:

The kerosene left after $1\text{feet}$

  $\begin{array}{l}=1-0.15\\ =0.85\end{array}$

The kerosene left after $2\text{feet}$

$0.85\times$ Kerosene left after $1\text{feet}$

  $\begin{array}{l}=0.85\times 0.85\\ ={\left(0.85\right)}^2\end{array}$

The kerosene left after $3\text{feet}$

$0.85\times$ Kerosene left after $2\text{feet}$

  $\begin{array}{l}=0.85\times 0.85\times 0.85\\ ={\left(0.85\right)}^3\end{array}$

If the impurity are completely removed

  $y=0$

The equation is

  $0={\left(0.85\right)}^x$

Has a least one solution

  ${\left(0.85\right)}^x>0$

Therefore the equation has no solution and the impurity never be completely removed.