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 293: 28

Answer

The expression for the area enclosed by the track as a function of radius $r$ of the track is $A\left( r \right)=440r$.

Work Step by Step

The total length of the path would be the sum of the length of straight paths and circumference of two semi circles. Write the expression for the total length of track. $2x+2\left( \pi r \right)=880$ Calculate $x$ in terms of r. $\begin{align} & 2x+2\left( \pi r \right)=880 \\ & 2x+2\pi r=880 \\ & 2x=880-2\pi r \\ & x=\frac{880-2\pi r}{2} \end{align}$ Solve further, $\begin{align} & x=\frac{2\left( 440-\pi r \right)}{2} \\ & =440-\pi r \end{align}$ The area enclosed by the track is the sum of the area of rectangle of sides x and $2r$ and semicircle of radius $r$. Write the expression of the area enclosed by the track. $\begin{align} & A=x\left( r \right)+2\left( \frac{1}{2}\pi {{r}^{2}} \right) \\ & =xr+\pi {{r}^{2}} \end{align}$ Substitute $440-\pi r$ for $x$. $\begin{align} & A=\left( 440-\pi r \right)r+\pi {{r}^{2}} \\ & =440r-\pi {{r}^{2}}+\pi {{r}^{2}} \\ & =440r \end{align}$ The area is function of r only which can also be expressed as, $A\left( r \right)=440r$ Hence, expression for the area enclosed by the track as a function of radius $r$ of the track is $A\left( r \right)=440r$.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.