Chapter 7.3, Problem 97E

To determine

To Find: Partial fraction decomposition of the rational function

Answer to Problem 97E

the partial fraction decomposition is

  $\frac{120p}{10,000-p^2}=\frac{60}{\left(100-p\right)}+\frac{-60}{\left(100+p\right)}$

Explanation of Solution

Given information: The magnitude of the range $C$ of exhaust temperatures

  $C=\frac{120p}{10,000-p^2}$

Concept used: Partial fraction decomposition is algebraic techniques for determining the constants in numerators

Calculation: Write the form of decomposition as follows

  $\frac{120p}{10,000-p^2}=\frac{60}{\left(100-p\right)}+\frac{-60}{\left(100+p\right)}=\frac{A}{\left(100-p\right)}+\frac{B}{\left(100+p\right)}$

Multiplying each side of this equation by the least common denominator

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

This equation is true for all $\left(p\right)$

Let

  $\begin{array}{l}p=-100 \\ -120\left(100\right)=0A+B\left(200\right) \\ B=-60\end{array}$

Again let $p=100$

  $\begin{array}{l}120\left(100\right)=200A+B\left(0\right) \\ A=60 \end{array}$

So, the partial fraction decomposition is

  $\frac{120p}{10,000-p^2}=\frac{60}{\left(100-p\right)}+\frac{-60}{\left(100+p\right)}$

  $\begin{array}{l}{y}_1=\frac{120p}{10,000-p^2}\\ y_2=\frac{60}{\left(100-p\right)}\\ y_3=\frac{-60}{\left(100+p\right)}\end{array}$

X	$y$1	$y$2	$y$ 3
100	error	error	-0.3
101	-60.3	-60	-0.29
102	-30.3	-30	-0.29702
103	-20.3	-20	-0.29556