Hello, Greenbaumenom!
Have you never done a Linear Programming problem?
A company makes and sells two types of 5ounce mixed nuts packages.
A lowgrade mixture containing 3 ounces of cashews and 2 ounces of peanuts
and a highgrade mixture containing 4 ounces of cashews and 1 ounce of peanuts.
The profit is $0.70 on each package of the lowgrade mixture
and $0.80 on each package of the highgrade mixture.
The mixing machine that gets filled every hour can hold at most 36 ounces of cashews
and no more than 14 ounces of peanuts.
By graphing, find:
a) the maximum hourly profit
b) the production schedule that maximizes the hourly profit. Why do they always ask these questions backwards ?? Organize the data . . .
. . $\displaystyle \begin{array}{ccc}
& \text{cashews} & \text{peanuts} \\ \hline
\text{Low }(x) & 3 & 2 \\
\text{High }(y) & 4 & 1 \\ \hline
\text{Total} & 36 & 14 \end{array}$
We have: .$\displaystyle \begin{array}{cc} x \:\geq\: 0 & {\color{blue}[1]}\\ y \: \geq\: 0 & {\color{blue}[2]}\\
3x + 4y \:\leq\: 36 & {\color{blue}[3]}\\ 2x + y \:\leq\: 14 & {\color{blue}[4]}\end{array}$
Profit Function: .$\displaystyle P \:=\;0.70x + 0.80y$
[1] and [2] places us in Quadrant 1.
The line of [3] is: .$\displaystyle 3x+4y \:=\:36$
. . It has intercepts (12,0) and (0,9).
. . Graph the line and shade the region below the line.
The line of [4] is: .$\displaystyle 2x + y \:=\:14$
. . It has intercepts (7,0) and (0,14).
. . Graph the line and shade the region below the line.
The graph looks like this: Code:

14 *
*
 *
 *
9 o *
::* *
:::::o
::::::* *
:::::::* *
::::::::* *
  o     o    *  
 7 12
The vertices of the shaded region are: .$\displaystyle (0,0),\;(7,0),\;(0,14)$
. . and the intersection of the two lines: $\displaystyle (4,6)$
Test them in the Profit Function to see which produces maximum profit.