Chapter 7.3, Problem 44E

To determine

Three ways to modify the dimensions of lake to represent the new area by the three types of special product patterns.

Answer to Problem 44E

New area can be ${(x+a)}^2$ , ${(x-a)}^2$ and $(x+a)(x-a)$ , where $\text{'}x\text{'}$ is the actual side of square lake and $\text{'}a\text{'}$ is the change in the side.

Explanation of Solution

Given information: The original area of the lake (in $m^2$ ) is $x^2$ .

Calculation:

1. Let the side of the lake be increased by $\text{'}a\text{'}$ $meters$ .

Then, new side of the lake= $(x+a)$ $meters$

New area of lake (in $m^2$ ) $={(x+a)}^2$ $=x^2+2ax+a^2$

2. Let the side of the lake be decreased by $\text{'}a\text{'}$ $meters$ .

Then, new side of the lake= $(x-a)$ $meters$

New area of lake (in $m^2$ ) $={(x-a)}^2$ $=x^2-2ax+a^2$

3. Let one side of lake be increased by $\text{'}a\text{'}$ $meters$ and the adjacent side be decreased by $\text{'}a\text{'}$ $meters$ .

Then, the lake will take the shape of a rectangle.

So, the new sides of lake (in $meters$ ) are $(x+a)$ and $(x-a)$ .

New area of lake (in $m^2$ ) $=(x+a)(x-a)$ $=x^2-a^2$