Chapter 8.4, Problem 17PPE

To determine

To find an expression for the area of shaded region

Answer to Problem 17PPE

The area of shaded region is $5\left(2x+3\right)\text{sq}\text{.units}$ .

Explanation of Solution

Given information :

Shaded region is

  

Since inner square has a side of length $x-1\text{units}\text{.}$

  $\Rightarrow \text{area}\text{of}\text{inner}\text{square}={\left(x-1\right)}^2\mathrm{.......}(i)$

And outer square has a side of length $x+4\text{units}\text{.}$

  $\Rightarrow \text{area}\text{of}\text{outer}\text{square}={\left(x+4\right)}^2\mathrm{.......}(ii)$

Thus, the area of shaded region is ${\left(x+4\right)}^2-{\left(x-1\right)}^2$

  $\begin{array}{l}=x^2+8x+16-\left(x^2-2x+1\right)\\ =x^2+8x+16-x^2+2x-1\\ =10x+15\\ =5\left(2x+3\right)\end{array}$

Hence, the area of shaded region is $5\left(2x+3\right)\text{sq}\text{.units}$ .