Chapter 11.3, Problem 23P

The area of a rectangle is $x^2-x-2$ and the length of the rectangle is $x+1$.

1. Find the width of the rectangle.
2. Find the area of the rectangle if the width is 4 m.

To determine

To find the width of the rectangle

Answer to Problem 23P

The width of the rectangle are $x-2$

Explanation of Solution

Given information:

The area of a rectangle is $x^2-x-2$ and the length of the rectangle is $x+1$

Calculation:

The width of the rectangle can be obtained as

Let, width of the rectangle is w

Therefore,

  $\text{Area of rectangle}$

  $=\text{lenght}\times \text{width}$

Put the value of $\text{area}=x^2-x-2\text{ and lenght}=x+1$ on the above equation

  $\begin{array}{l}{x}^2-x-2=\left(x+1\right)w\\ w=\frac{x^2-x-2}{x+1}\\ w=\frac{x^2-\left(2x-x\right)-2}{x+1}\left[\text{write }x=\left(2x-x\right)\right]\\ w=\frac{x^2-2x+x-2}{x+1}\\ w=\frac{x\left(x-2\right)+1\left(x-2\right)}{x+1}\left[\begin{array}{l}\text{the first and second term has a common factor,}x\text{ and}\\ \text{the third and fourth term has a common factor,1}\end{array}\right]\\ w=\frac{\left(x-2\right)\left(x+1\right)}{x+1}\left[\text{divide numerator and denominator by }\left(x+1\right)\right]\\ w=x-2\end{array}$

Hence,

The width of the rectangle are $x-2$

To determine

To find the area of the rectangle

Answer to Problem 23P

The area of the rectangle are $28{\text{m}}^2$

Explanation of Solution

Given information:

The width is $4\text{m}$

Calculation:

The area of the rectangle can be obtained as

In previous part we find the width

  $\begin{array}{l}\text{width of the rectangle}=x-2\\ 4=x-2\left[\begin{array}{l}\text{width}=w\\ w=4\end{array}\right]\\ x=4+2\\ x=6\end{array}$

Therefore,

  $\begin{array}{l}\text{length of the rectangle}=x+1\\ l=6+1\left[\begin{array}{l}l=\text{length}\\ x=6\end{array}\right]\\ l=7\end{array}$

  $\text{Area of rectangle}$

  $=\text{lenght}\times \text{width}$

Put the value of $\text{width}=4\text{ and lenght}=7$ on above equation

  $\begin{array}{l}a=7\times 4\left[a=\text{area}\right]\\ a=28{\text{m}}^2\end{array}$

Hence,

The area of the rectangle are $28{\text{m}}^2$