Chapter 3.3, Problem 42E

To determine

To find: the area of region described by the inequalities.

Answer to Problem 42E

30

Explanation of Solution

Given information:

The inequalities are shown below.

  $\begin{array}{c}y\ge 2\left|x-3\right|+4\\ x-2y\ge -20\end{array}$

Calculation:

Plot the graph for both the inequalities and shade the common region enclosed by them.

  

The shaded area $\Delta ABC$ is enclosed by the given region and the vertices are $\left(0,10\right)\left(3,4\right)$ and $\left(8,14\right)$ .

Put the vertex coordinates of $A\text{ and }B$ in the slope formula to evaluate slope of $AB$ as follows:

  $\begin{array}{c}{m}_{AB}=\frac{y_2-y_1}{x_2-x_1}\\ =\frac{4-10}{3-0}\\ =\frac{-6}{3}\\ =-2\end{array}$

Put the vertex coordinates of $A\text{ and }C$ in the slope formula to evaluate slope of $AB$ as follows:

  $\begin{array}{c}{m}_{AC}=\frac{y_2-y_1}{x_2-x_1}\\ =\frac{14-10}{8-0}\\ =\frac{4}{8}\\ =\frac{1}{2}\end{array}$

Since the slope value of $AB$ and $AC$ are negative reciprocal of each other, so $AB\perp AC$ and

  $\Delta ABC$ is a right angled triangle.

Use the distance formula mentioned below to evaluate the length of $AB$ and $AC$ as follows:

  $d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

  $\begin{array}{c}{d}_{AB}=\sqrt{{\left(3-0\right)}^2+{\left(4-10\right)}^2}\left[\text{Let length of }AB=d_{AB}\right]\\ =\sqrt{{\left(3\right)}^2+{\left(-6\right)}^2}\\ =\sqrt{9+36}\\ =\sqrt{40}\\ =3\sqrt{5}\end{array}$

  $\begin{array}{c}{d}_{AC}=\sqrt{{\left(8-0\right)}^2+{\left(14-10\right)}^2}\left[\text{Let length of }AC=d_{AC}\right]\\ =\sqrt{{\left(8\right)}^2+{\left(4\right)}^2}\\ =\sqrt{64+16}\\ =\sqrt{80}\\ =4\sqrt{5}\end{array}$

The area of right angle triangle $\Delta ABC$ is evaluated as follows:

  $\begin{array}{c}\text{Area of }\Delta ABC=\frac{1}{2}\left(base\right)\left(height\right)\\ =\frac{1}{2}\left(d_{AB}\right)\left(d_{AC}\right)\\ =\frac{1}{2}\left(3\sqrt{5}\right)\left(4\sqrt{5}\right)\\ =30\end{array}$

Thus, the area enclosed by the inequalities is 30.