Precalculus (6th Edition) Blitzer

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

Chapter 1 - Section 1.10 - Modeling with Functions - Exercise Set - Page 294: 42

Answer

The combined lengths of roads from town $A$ to expressway and town $B$ to expressway, $f$ in terms of $x$, is, $\sqrt{36+{{x}^{2}}}+\sqrt{{{x}^{2}}-24x+153}$.

Work Step by Step

Consider the length of road from town $A$ to expressway as ${{h}_{1}}$ and the length of road from town $B$ to expressway as ${{h}_{2}}$ According to the Pythagoras theorem In a right-angle triangle, ${{h}^{2}}={{p}^{2}}+{{b}^{2}}$ …… (1) Substitute $6$ for $p$ , $x$ for $b$ and ${{h}_{1}}$ for $h$ in equation (1) $\begin{align} & {{h}_{1}}=\sqrt{{{6}^{2}}+{{x}^{2}}} \\ & =\sqrt{36+{{x}^{2}}} \end{align}$ The length of road from town $A$ to expressway ${{h}_{1}}=\sqrt{36+{{x}^{2}}}$ Substitute $3$ for $p$ , $12-x$ for $b$ and ${{h}_{2}}$ for $h$ in equation (1) ${{h}_{2}}=\sqrt{{{3}^{2}}+{{\left( 12-x \right)}^{2}}}$ Use formula ${{\left( A-B \right)}^{2}}={{A}^{2}}-2AB+{{B}^{2}}$ in the above equation $\begin{align} & {{h}_{2}}=\sqrt{9+144-24x+{{x}^{2}}} \\ & =\sqrt{153-24x+{{x}^{2}}} \end{align}$ Or ${{h}_{2}}=\sqrt{{{x}^{2}}-24x+153}$ The length of road from town $B$ to expressway ${{h}_{2}}=\sqrt{{{x}^{2}}-24x+153}$ The combined lengths of roads from town $A$ to expressway and town $B$ to expressway $f={{h}_{1}}+{{h}_{2}}$ Substitute $\sqrt{36+{{x}^{2}}}$ for ${{h}_{1}}$ and $\sqrt{{{x}^{2}}-24x+153}$ for ${{h}_{2}}$ in the above equation $f=\sqrt{36+{{x}^{2}}}+\sqrt{{{x}^{2}}-24x+153}$ The required combined lengths of roads from town $A$ to expressway and town $B$ to expressway, $f$ in terms of $x$ , where the distance of towns $A$ and $B$ from expressway are $6\text{ and }3\text{ miles}$ respectively and towns are $12\text{ miles}$ apart from each other, is, $\sqrt{36+{{x}^{2}}}+\sqrt{{{x}^{2}}-24x+153}$.
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