Chapter 3.4, Problem 30HP

To determine

To calculate: The area of bounded region formed by constraints $y\ge \left|x\right|-3,y\le -\left|x\right|+3\text{ and }x\ge \left|y\right|$.

Answer to Problem 30HP

The area of bounded region is $4.5\text{ square units}$.

Explanation of Solution

Given information:

The constraints $y\ge \left|x\right|-3,y\le -\left|x\right|+3\text{ and }x\ge \left|y\right|$.

Formula used:

The area of rhombus is the product of length of its diagonals divided by 2.

Calculation:

Consider the constraints $y\ge \left|x\right|-3,y\le -\left|x\right|+3\text{ and }x\ge \left|y\right|$.

Plot the constraints and compute the common region.

  

The vertices of the common region are $\left(0,0\right),\left(1.5,1.5\right),\left(3,0\right)\text{ and }\left(1.5,-1.5\right)$.

It form a rhombus.

Recall that the area of rhombus is the product of length of its diagonals divided by 2.

Here length of each diagonal is 3.

Therefore, area of common region is,

  $\begin{array}{c}A=\frac{3\times 3}{2}\\ =4.5\end{array}$

Thus, the area of bounded region is $4.5\text{ square units}$.