Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter P - Section P.6 - Rational Expressions - Exercise Set - Page 86: 96

Answer

The required solution is $\frac{4{{x}^{2}}+22x}{\left( x+5 \right)\left( x+6 \right)}$

Work Step by Step

We know that the perimeter of the rectangle can be calculated by using the formula: $P=2l+2b$ Here, $l$ refers to the length and $b$ refers to the width of the rectangle. Put $l=\frac{x}{x+5}$ and $b=\frac{x}{x+6}$ in the equation of $P$. $\begin{align} & P=2\left( \frac{x}{x+5} \right)+2\left( \frac{x}{x+6} \right) \\ & =\frac{2x}{x+5}+\frac{2x}{x+6} \end{align}$ And solve the above rational expression by taking the least common denominator. $\begin{align} & P=\frac{2x}{x+5}\times \frac{\left( x+6 \right)}{\left( x+6 \right)}+\frac{2x}{x+6}\times \frac{\left( x+5 \right)}{\left( x+5 \right)} \\ & =\frac{2x\left( x+6 \right)}{\left( x+5 \right)\left( x+6 \right)}+\frac{2x\left( x+5 \right)}{\left( x+6 \right)\left( x+5 \right)} \\ & =\frac{2{{x}^{2}}+12x+2{{x}^{2}}+10x}{\left( x+5 \right)\left( x+6 \right)} \\ & =\frac{4{{x}^{2}}+22x}{\left( x+5 \right)\left( x+6 \right)} \end{align}$ Hence, the perimeter of the rectangle as a single rational expression is $\frac{4{{x}^{2}}+22x}{\left( x+5 \right)\left( x+6 \right)}$.
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