Chapter 8, Problem 37SGR

To determine

To write an expression of the shaded region.

Answer to Problem 37SGR

  $3x^2-21$

Explanation of Solution

Given:

  

Formula used:

Area of rectangle $=l\times b$

From the given figure, there are two rectangles, the area of the shaded region is between the rectangles.

Finding the area of both the rectangle and finding the difference of the area of the rectangles gives the area of the shaded region.

Dimensions of the outer rectangle:

  $\begin{array}{l}l=2x-5\\ b=2x-5\end{array}$

  $\begin{array}{c}\text{Area of the outer rectangle}=(2x-5)\times (2x+5)\\ $ ={(2x)}^2-5^2\text{ [}\because \text{(}a\text{+}b\text{)(}a-b\text{)=}{a}^2-b^2\text{]}$=4x^2-25\end{array}$

Dimensions of the inner rectangle:

  $\begin{array}{l}l=x+2\\ b=x-2\end{array}$

  $\begin{array}{c}\text{Area of the inner rectangle}=(x+2)\times (x-2)\\ $ ={(x)}^2-2^2\text{ [}\because \text{(}a\text{+}b\text{)(}a-b\text{)=}{a}^2-b^2\text{]}$=x^2-4\end{array}$

Difference of the areas of the outer and inner rectangles gives the area of the shaded region.

  $\begin{array}{c}\text{Difference in the areas of the rectangles}=(4x^2-25)-(x^2-4)\\ =(4x^2-x^2\text{) +(}-25+4\text{) }\\ =3x^2-21\end{array}$

Conclusion:

Therefore, the area of the shaded region is $3x^2-21.$