Chapter 3.4, Problem 50E

a.

To determine

To find: the inequalities for the given situation.

Answer to Problem 50E

  $\begin{array}{l}4a+b\le 32\\ a+6b\le 54\\ a\ge 0\\ b\ge 0\\ G\left(a,b\right)=a+b\end{array}$

Explanation of Solution

Given information:

Let number of gallons of color $A\text{ and }B$ be $a\text{ and }b$ respectively.The table below shown below represents the possible combinations of each color dye.

	Blue Dye	Red Dye
Total Dye	32	54
Color   $A$	4	1
Color   $B$	1	6

Calculation:

Assume, $a\ge 0\text{ and }b\ge 0$ . The inequalities describing the situation are as follows:

  $\begin{array}{l}4a+b\le 32\\ a+6b\le 54\end{array}$

The maximum function will be as follows:

  $G\left(a,b\right)=a+b$

b.

To determine

To find: the maximum number of gallons possible.

Answer to Problem 50E

14 gallons

Explanation of Solution

Given information:

Let number of gallons of color $A\text{ and }B$ be $a\text{ and }b$ respectively. The inequalities and maximum function evaluated in part a. are as follows:

  $\begin{array}{l}4a+b\le 32\\ a+6b\le 54\\ a\ge 0\\ b\ge 0\\ G\left(a,b\right)=a+b\end{array}$

Calculation:

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

  

The vertices are $\left(0,9\right),\left(6,8\right)\text{,}\left(8,0\right)\text{ and }\left(0,0\right)$ . Now test each vertex as shown below.

  $\begin{array}{c}G\left(0,9\right)=0+9\\ =9\\ G\left(6,8\right)=6+8\\ =14\\ G\left(8,0\right)=8+0\\ =8\\ G\left(0,0\right)=0+0\\ =0\end{array}$

Thus, the maximum number of gallons possible is 14.