Chapter 14.8, Problem 7CYU

(a)

To determine

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

Answer to Problem 7CYU

The area of the shaded region in factored form is $\left(x+3\right)\left(x-8\right)$ .

Explanation of Solution

Given information:

The dimensions of larger rectangle are $\text{Length}=x$ and $\text{breadth}= x-5$ and the dimensionof Small rectangleare $\text{Length}=6\text{ and breadth}=4$.

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 $\text{Length}=x$ and $\text{breadth}= x-5$ and the dimensionof Small rectangleare $\text{Length}=6\text{ and breadth}=4$.

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

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

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

Substitute $\text{6 for length}$ and $\text{4 for breadth}$ to obtain the Area $\left(A_1\right)$ of small rectangle.

  $\begin{array}{c}\left(A_2\right)=6\cdot 4\\ =24\end{array}$

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

  $\begin{array}{c}\left(A_1\right)-\left(A_2\right)=x^2-5x-24\\ =x^2-8x+3x-24\\ =x\left(x-8\right)+3\left(x-8\right)\\ =\left(x+3\right)\left(x-8\right)\end{array}$

Therefore,the area of the shaded region in factored form is $\left(x+3\right)\left(x-8\right)$ .

Conclusion:

Thus, the area of the shaded region in factored form is $\left(x+3\right)\left(x-8\right)$ .

(b)

To determine

To find: The area of the shaded region if value of $x$ is 10.

Answer to Problem 7CYU

The area of the shaded region is $26$ if $x=10$ .

Explanation of Solution

Given information:

The value of $x\text{ is 10}$.

Calculation:

It is given that $x=10$ .

Substitute $10$ for $x$ in the obtained expression $\left(x+3\right)\left(x-8\right)$ from part (a) to obtain area of the shaded region if value of $x$ is 10 .

  $\begin{array}{c}\left(10+3\right)\left(10-8\right)=\left(13\right)\left(2\right)\\ =26\end{array}$

Thereforethe area of the shaded region is $26$ if $x=10$ .

Conclusion:

Thus,the area of the shaded region is $26$ if $x=10$ .