Chapter 2.7, Problem

To determine

To find: the area enclosed by the polygonal convex set.

Answer

70.93 units

Explanation of Solution

Given information:

  $\begin{array}{c}y\ge 0\\ x\le 12\\ 2x+6y\le 84\\ 2x-3y\le -3\\ 8x+3y\ge 33\end{array}$

Calculation:

Use a graphing calculator to plot the inequalities and find the vertices as shown below.

  

  $\begin{array}{l}A=\left(\frac{-9}{7},\frac{101}{7}\right)\\ B=\left(13,\frac{87}{9}\right)\\ C=\left(3,3\right)\\ D=\left(12,10\right)\\ E=\left(12,9\right)\end{array}$

Now create a matrix of larger triangle $\Delta ABC$ and smaller triangle $\Delta BDE$ using $z=1$ for each as shown below.

  $\begin{array}{l}\Delta ABC=\left[\begin{array}{ccc}\begin{array}{l}-\raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$7$}\right.\\ 13\\ 3\end{array}& \begin{array}{l}\raisebox{1ex}{$107$}\!\left/ \!\raisebox{-1ex}{$7$}\right.\\ \raisebox{1ex}{$87$}\!\left/ \!\raisebox{-1ex}{$9$}\right.\\ 3\end{array}& \begin{array}{l}1\\ 1\\ 1\end{array}\end{array}\right]\\ \Delta BDE=\left[\begin{array}{ccc}\begin{array}{l}12\\ 13\\ 12\end{array}& \begin{array}{l}10\\ \raisebox{1ex}{$87$}\!\left/ \!\raisebox{-1ex}{$9$}\right.\\ 9\end{array}& \begin{array}{l}1\\ 1\\ 1\end{array}\end{array}\right]\end{array}$

The area of triangles can be evaluated by evaluating the half value of their determinants as shown below.

  $\begin{array}{c}\text{Area of }\Delta ABC=\frac{1}{2}{\det }_{\Delta ABC}\\ =\frac{1}{2}\left[-\raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$7$}\right.\left\{\left(\raisebox{1ex}{$87$}\!\left/ \!\raisebox{-1ex}{$9$}\right.\times 1\right)-\left(3\times 1\right)\right\}-\raisebox{1ex}{$107$}\!\left/ \!\raisebox{-1ex}{$7$}\right.\left\{\left(13\times 1\right)-\left(3\times 1\right)\right\}+1\left\{\left(13\times 3\right)-\left(\raisebox{1ex}{$87$}\!\left/ \!\raisebox{-1ex}{$9$}\right.\times 3\right)\right\}\right]\\ =\frac{1}{2}\left(\frac{1000}{7}\right)\\ =71.43\text{ units}\end{array}$

  $\begin{array}{c}\text{Area of }\Delta BDE=\frac{1}{2}{\det }_{\Delta BDE}\\ =\frac{1}{2}\left[12\left\{\left(\raisebox{1ex}{$87$}\!\left/ \!\raisebox{-1ex}{$9$}\right.\times 1\right)-\left(3\times 1\right)\right\}-10\left\{\left(13\times 1\right)-\left(3\times 1\right)\right\}+1\left\{\left(13\times 9\right)-\left(\raisebox{1ex}{$87$}\!\left/ \!\raisebox{-1ex}{$9$}\right.\times 12\right)\right\}\right]\\ =\frac{1}{2}\left(1\right)\\ =0.5\text{ units}\end{array}$

The area enclosed by polygonal convex set can be evaluated as follows:

  $\begin{array}{c}\text{Area enclosed}=\Delta ABC-\Delta BDE\\ =71.43-0.5\\ =70.93\text{ units}\end{array}$

Thus, the area enclosed by the polygonal convex set is 70.93 units.