Chapter 14.8, Problem 18PPS

(a)

To determine

To find: Thearea of the shaded region in factored form.

Answer to Problem 18PPS

The area of the shaded region in factored form is. $\left(l+1\right)\left(l-6\right)$ .

Explanation of Solution

Given information:

The dimensions of larger rectangle are $\text{Length}=l$ and $\text{breadth}= l-5$ and the dimensionof Small rectangleare $\text{Length}=3\text{ and breadth}=2$.

Formula used:

The area of the rectangle is as follows:

  $\text{Area of rectangle}=\text{length}\cdot \text{breath}$ .

Calculation:

As it is given that the dimensions of larger rectangle $\text{Length}=l$ and $\text{breadth}= l-5$ and the dimensionof Small rectangleare $\text{Length}=3\text{ and breadth}=2$.

Substitute $x\text{ for length}$ and $x-5$

  $\text{for breadth}$ to obtain the Area $\left(A_1\right)$ of big rectangle.

  $\begin{array}{c}\left(A_1\right)=\left(l\right)\cdot \left(l-5\right)\\ =l^2-5l\end{array}$

Substitute $\text{3 for length}$ and $\text{2 for breadth}$ to obtain the Area $\left(A_1\right)$ of small rectangle.

  $\begin{array}{c}\left(A_2\right)=3\cdot 2\\ =6\end{array}$

Subtract the Area of rectangle $\left(A_2\right)$ from Area of rectangle $\left(A_1\right)$ to obtain the area of shaded region.

  $\begin{array}{c}\left(A_1\right)-\left(A_2\right)=l^2-5l-6\\ =l^2-6l+l-6\\ =l\left(l-6\right)+1\left(l-6\right)\\ =\left(l+1\right)\left(l-6\right)\end{array}$

Therefore,the area of the shaded region in factored form is $\left(l+1\right)\left(l-6\right)$ .

Conclusion:

Thus, the area of the shaded region in factored form is $\left(l+1\right)\left(l-6\right)$ .

(b)

To determine

To find: The area of the shaded region if value of $l$ is $8$ .

Answer to Problem 18PPS

The area of the shaded region is $18$ if $l=8$ .

Explanation of Solution

Given information:

The value of $l$ is $8$ .

Calculation:

It is given that $l=8$ .

Substitute $10$ for $x$ in the obtained expression $\left(l+1\right)\left(l-6\right)$ from part (a) to obtain area of the shaded region if value of $l$ is $8$ .

  $\begin{array}{c}\left(8+1\right)\left(8-6\right)=\left(9\right)\left(2\right)\\ =18\end{array}$

Thereforethe area of the shaded region is $18$ if $l=8$ .

Conclusion:

Thus,the area of the shaded region is $18$ if $l=8$ .