Chapter 7.7, Problem 5P

To determine

Tofind: thegreatest possible area of the rectangle.

Answer to Problem 5P

The greatest possible area of the rectangle is $625{\text{cm}}^2$ .

Explanation of Solution

Given information:

The perimeter of rectangle is $100\text{cm}$ .

Calculation: to find the greatest area, let $x$ represents the length of the rectangle, then the width of the rectangle,

$\begin{array}{l}2\left(x+b\right)=100\\ x+b=50\text{ [divide]}\\ b=50-x\text{ [subtract]}\end{array}$

So the area of the rectangle is,

$\begin{array}{l}a=x\left(50-x\right)\\ a=50x-x^2\text{ [multiply]}\end{array}$

To find the greatest area , first differentiate and then equal to zero,

$\begin{array}{l}a=50x-x^2\\ a\text{'}=50-2x\text{ [differentiate]}\\ 50-2x=0\text{ [equal to zero]}\\ x=25\text{ [simplify]}\end{array}$

Now, substitute in the area function,

$\begin{array}{l}a=50\left(25\right)-{\left(25\right)}^2\text{ [substitute]}\\ a=1250-625\text{ [multiply]}\\ a=625\text{ [subtract]}\end{array}$

Thus, the greatest possible area is $625{\text{cm}}^2$ .