# Hard Math Question!

• May 16th 2008, 09:53 AM
Greenbaumenom
Hard Math Question!
The owner os a small Nutt's & Nuts company makes and sells two types of 5-ounce mixed nuts packages. A low-grade mixture containing 3 ounces of cashews and 2 ounces of peanuts and a high-grade mixture containing 4 ounces of cashews and 1 ounce of peanuts. The profit is $.70 on each package of the low-grade mixture and$.80 on each package of the high-grade mixture. The mixing machine that gets filled every hour can hold utmost 36 ounces of cashews and no more than 14 ounces of peanuts. By graphing, find: a) the maximum hourly profit, and b) the production schedule that maximizes the hourly profit.

• May 16th 2008, 11:48 AM
Soroban
Hello, Greenbaumenom!

Have you never done a Linear Programming problem?

Quote:

A company makes and sells two types of 5-ounce mixed nuts packages.
A low-grade mixture containing 3 ounces of cashews and 2 ounces of peanuts
and a high-grade mixture containing 4 ounces of cashews and 1 ounce of peanuts.

The profit is $0.70 on each package of the low-grade mixture and$0.80 on each package of the high-grade 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 . . .

. . $\displaystyle \begin{array}{c|c|c|} & \text{cashews} & \text{peanuts} \\ \hline \text{Low }(x) & 3 & 2 \\ \text{High }(y) & 4 & 1 \\ \hline \text{Total} & 36 & 14 \end{array}$

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.