Chapter 7.3, Problem 62E

To determine

To calculate: the partial fraction decomposition for the rational functions and also verify result by using a table comparing the original function with the partial functions.

Answer to Problem 62E

The partial fraction equation is $\frac{120p}{10000-p^2}=\frac{-60}{100+p}+\frac{60}{100-p}$

Explanation of Solution

Given information:

The predicted cost C for a company to remove p% of a chemical from its waste water is given by the model

  $C=\frac{120p}{10000-p^2},0\le p\le 100$

Calculation:

From the given equation, we can decompose it partially like,

  $\begin{array}{l}C=\frac{120p}{10000-p^2},0\le p\le 100\\ \frac{120p}{10000-p^2}=\frac{A}{100+p}+\frac{B}{100-p}\end{array}$

Here A and B are the variables whose value is to be calculated.

Now by solving it, we get,

  $120p=A\left(100-p\right)+B\left(100+p\right)$

Let us put $p=100$ in the above equation, we get,

  $\begin{array}{l}120\left(100\right)=A\left(100-100\right)+B\left(100+100\right)\\ 12000=B\left(200\right)\\ B=60\end{array}$

Let us put $p=0$ in the above equation, we get,

  $\begin{array}{l}120\left(0\right)=A\left(100\right)+B\left(100\right)\\ A+B=0\\ A+60=0\left(\because B=60\right)\\ A=-60\end{array}$

The partial fraction equation is $\frac{120p}{10000-p^2}=\frac{-60}{100+p}+\frac{60}{100-p}$

The table which is obtained from the calculator for different values of P is as follows.

  $\begin{array}{ccc}P& \frac{120p}{10000-p^2}& \frac{60}{100-p}-\frac{60}{100+p}\\ 97& 19.695& 19.695\\ 98& 29.697& 29.697\\ 99& 59.698& 59.698\\ 100& \text{ERROR}& \text{ERROR}\\ 101& -60.3& -60.3\\ 102& -30.3& -30.3\\ 103& -20.3& -20.3\end{array}$

Here for different P values, the returned by the partial fraction and the provided fraction are same. Hence the partial fraction is correct.