Chapter 5, Problem 45RE

To determine

To find the maximum area.

Answer to Problem 45RE

The maximum area is $50.$

Explanation of Solution

Given information: Given perimeter $100.$

Calculation:

Given perimeter $2r+s=100$

  $r=50-\frac{s}{2}$

  $\begin{array}{l}r=50-\frac{2\pi r\theta }{2}\text{ Since }s=2\pi r\theta \\ r=50-\pi r\theta \\ \theta =\frac{50-r}{\pi r}\end{array}$

Area $A=\pi r^2\theta$

  $\begin{array}{l}A=\pi r^2\left(\frac{50-r}{\pi r}\right)\text{ Since }\theta =\frac{50-r}{\pi r}\\ A=50r-r^2\end{array}$

  $\begin{array}{l}{A}^{\prime }=50-2r\\ A^{\prime }=0\\ 50-2r=0\\ r=25\end{array}$

  ${A^{\prime }}^{\prime }=-2<0$

Thus area is maximum when $r=25$

  $s=2\pi r\theta$

  $\begin{array}{l}s=2\left(50-r\right)\\ s=2\left(50-25\right)\\ s=50\end{array}$

Thus the maximum area is $50.$