Chapter 8.3, Problem 34PPS

To determine

To find: The expression which represents the area of the shaded region.

Answer to Problem 34PPS

  $24x^2-\frac{3}{2}$

Explanation of Solution

Given:

  

Calculation:

In the given figure, the length of the rectangle is $5x+2x-3=7x-3$

The height of the triangle is $4x+1$ and that of base is $2x-3$

The area of the rectangle is given by

  $\begin{array}{c}{\mathrm{A }}_r=\left(7x-3\right)\left(4x+1\right)\\ =28x^2+7x-12x-3\\ =28x^2-5x-3\end{array}$

And the area of the triangle is given by

  $\begin{array}{c}{\mathrm{A }}_t=\frac{1}{2}\cdot \left(4x+1\right)\cdot \left(2x-3\right)1\\ =\frac{1}{2}\left(8x^2+2x-12x-3\right)\\ =\frac{1}{2}\left(8x^2-10x-3\right)\\ =4x^2-5x-\frac{3}{2}\end{array}$

Therefore, the area of the shaded region is given by

  $\begin{array}{l}{\mathrm{A }}_r-{\mathrm{A }}_t\\ =28x^2-5x-3-4x^2+5x+\frac{3}{2}\\ =24x^2-\frac{3}{2}\end{array}$

Therefore, the expression that represents the area of the shaded region is $24x^2-\frac{3}{2}$