Chapter 2.6, Problem 25E

a.

To determine

To find: the coordinates of each vertex.

Answer to Problem 25E

  $\left(\frac{11}{2},0\right),\left(\frac{13}{2},0\right),\left(\frac{5}{2},\frac{17}{2}\right),\left(\frac{29}{3},\frac{19}{3}\right),\left(\frac{15}{2},\frac{17}{2}\right),\left(\frac{5}{2},2\right)\text{ and}\left(\frac{6}{5},\frac{23}{5}\right)$

Explanation of Solution

Given information:

The system of inequalities is shown below.

  $\begin{array}{cccc}\begin{array}{l}y\le 16-x\\ 0\le 2y\le 17\end{array}& \begin{array}{l}3y\ge -2x+11\\ y\le 3x+1\end{array}& \begin{array}{l}y\ge 2x-13\\ y\ge 7-2x\end{array}& \end{array}$

Calculation:

Graph for the given inequalities is shown below.

  

The vertices of the polygon boundary are $\left(\frac{11}{2},0\right),\left(\frac{13}{2},0\right),\left(\frac{5}{2},\frac{17}{2}\right),\left(\frac{29}{3},\frac{19}{3}\right),\left(\frac{15}{2},\frac{17}{2}\right),\left(\frac{5}{2},2\right)$ and $\left(\frac{6}{5},\frac{23}{5}\right)$ .

b.

To determine

To find: the maximum and minimum values for $f\left(x,y\right)$ .

Answer to Problem 25E

Maximum value is 88.5.

Minimum value is $24.5$ .

Explanation of Solution

Given information:

  $f\left(x,y\right)=5x+6y$

The vertices are $\left(\frac{11}{2},0\right),\left(\frac{13}{2},0\right),\left(\frac{5}{2},\frac{17}{2}\right),\left(\frac{29}{3},\frac{19}{3}\right),\left(\frac{15}{2},\frac{17}{2}\right),\left(\frac{5}{2},2\right)$ and $\left(\frac{6}{5},\frac{23}{5}\right)$ .

Calculation:

The maximum and minimum value of $f\left(x,y\right)$ will occur at the given points. Solve the function to get maximum and minimum as follows:

  $\begin{array}{c}f\left(\frac{11}{2},0\right)=5\left(\frac{11}{2}\right)+6\left(0\right)\\ =\frac{55}{2}\\ =27.5\\ f\left(\frac{13}{2},0\right)=5\left(\frac{13}{2}\right)+6\left(0\right)\\ =\frac{65}{2}\\ =32.5\\ f\left(\frac{5}{2},\frac{17}{2}\right)=5\left(\frac{5}{2}\right)+6\left(\frac{17}{2}\right)\\ =\frac{127}{2}\\ =63.5\\ f\left(\frac{29}{3},\frac{19}{3}\right)=5\left(\frac{29}{3}\right)+6\left(\frac{19}{3}\right)\\ =\frac{259}{3}\\ =86.3\\ f\left(\frac{15}{2},\frac{17}{2}\right)=5\left(\frac{15}{2}\right)+6\left(\frac{17}{2}\right)\\ =\frac{177}{2}\\ =88.5\\ f\left(\frac{5}{2},2\right)=5\left(\frac{5}{2}\right)+6\left(2\right)\\ =\frac{49}{2}\\ =24.5\\ f\left(\frac{6}{5},\frac{23}{5}\right)=5\left(\frac{6}{5}\right)+6\left(\frac{23}{5}\right)\\ =\frac{168}{2}\\ =33.6\end{array}$

Thus, the maximum and minimum value of $f\left(x,y\right)$ is 88.5 and 24.5 respectively.