Chapter 7, Problem 59RE

Solution

To calculate: The solution of the system of inequality and given he coordinates of any corner points.

  $\begin{array}{c}4x+9y\le 360\\ x+4y\le 360\\ x+y\ge 90\end{array}$

The common region in the graph is the solution of the inequality.

Given Information:

The system of inequality is,

  $\begin{array}{c}4x+9y\le 360\\ x+4y\le 360\\ x+y\ge 90\end{array}$

Calculation:

Consider the given system of inequality,

  $\begin{array}{c}4x+9y\le 360\\ x+4y\le 360\\ x+y\ge 90\end{array}$

The boundary point is included if the inequality containing $\le \text{ or }\ge$ symbols and the boundary point is excluded if the symbols $<\text{ or }>$  is used.

In the first inequality, the boundary is $4x+9y=360$ , which is slanted line. Find the intercept of the line. Substitute 0 for x and then 0 for y in the line one by one.

  $\begin{array}{c}4x+9\left(0\right)=360\\ 4x=360\\ x=90\end{array}$

And,

  $\begin{array}{c}4\left(0\right)+9y=360\\ 9y=360\\ y=40\end{array}$

The intercepts of the equation are $\left(90,0\right)\text{ and }\left(0,40\right)$ .

To determine the half-plane to be shaded, use a test point not on the boundary, say  $\left(0,0\right)$ . Then,

  $\begin{array}{c}4\left(0\right)+9\left(0\right)\le 360\\ 0\le 360\end{array}$

The statement is true so the shade the half-plane where  $\left(0,0\right)$  is located.

In the second inequality, the boundary is $x+4y=360$ , which is slanted line. Find the intercept of the line. Substitute 0 for x and then 0 for y in the line one by one.

  $\begin{array}{c}x+4\left(0\right)=360\\ 9x=360\\ x=40\end{array}$

And,

  $\begin{array}{c}\left(0\right)+4y=360\\ 4y=360\\ y=90\end{array}$

The intercepts of the equation are $\left(40,0\right)\text{ and }\left(0,90\right)$ .

To determine the half-plane to be shaded, use a test point not on the boundary, say  $\left(0,0\right)$ . Then,

  $\begin{array}{c}0+4\left(0\right)\le 360\\ 0\le 360\end{array}$

The statement is true so the shade the half-plane where  $\left(0,0\right)$  is located.

In the third inequality, the boundary is $x+y=90$ , which is slanted line. Find the intercept of the line. Substitute 0 for x and then 0 for y in the line one by one.

  $\begin{array}{c}x+0=90\\ x=90\end{array}$

And,

  $\begin{array}{c}0+y=90\\ y=90\end{array}$

The intercepts of the equation are $\left(90,0\right)\text{ and }\left(0,90\right)$ .

To determine the half-plane to be shaded, use a test point not on the boundary, say  $\left(0,0\right)$ . Then,

  $\begin{array}{c}0+0\ge 90\\ 0\ge 30\end{array}$

The statement is not true so the shade the half-plane where  $\left(0,0\right)$  is not located.

Now, make the graph of the inequality.

  

Thus, the shaded part by both the inequality is the solution of the inequality.

And the corner points are defined as,

  $\left(0,90\right),\left(90,0\right),\text{ and }\left(27.692,27.692\right)$ .

Hence, the corner points are $\left(0,90\right),\left(90,0\right),\text{ and }\left(27.692,27.692\right)$ .