# Inequalities help...

• Feb 10th 2008, 04:47 AM
struck
Inequalities help...
This question has to be solved using inequalities with the means of a graph. But despite breaking my head on it several times, I am unable to reach a viable solution.

"A supermarket sells two kinds of washing powder, Sure-clean and Quick-Wash. At least three times as much Sure-Clean is sold as Quick-wash. The supermarket has, at most, 1800cm^2 of shelf space for washing powders. A box of Sure-Clean requires 25cm^2 of shelf space, whilst a box of Quick-wash requires 15cm^2. The profit per box is 8p for Sure-Clean and 12p for Quick-Wash. How many boxes of each kind of powder should be stocked for maximum profit? What is the maximum profit?"

Please help me setup the inequality. I can handle the graphing and finding the solution for the inequality.

Thank you :)
• Feb 10th 2008, 05:36 AM
Soroban
Hello, struck!

Quote:

A supermarket sells two kinds of washing powder, Sure-Clean and Quick-Wash.
At least three times as much Sure-Clean is sold as Quick-Wash.
The supermarket has, at most, 1800cm² of shelf space for washing powders.
A box of Sure-Clean needs 25cm² of shelf space; a box of Quick-wash needs 15cm².
The profit per box is 8p for Sure-Clean and 12p for Quick-Wash.
How many boxes of each kind of powder should be stocked for maximum profit?
What is the maximum profit?

We have: .$\displaystyle \begin{array}{c|c} & \text{space} \\ \hline \text{Sure (x)} & 25 \\ \text{Quick (y)} & 15 \\ \hline \text{Total} & 1800 \end{array}\quad\Rightarrow\quad 25x + 15y \:\leq \:1800$

We are also told that: .$\displaystyle x \:\geq \:3y\quad \Rightarrow\quad x -3y \:\geq \:0$

And we have: .$\displaystyle P \;=\;8x+12y$

. . Go for it!

• Feb 13th 2008, 04:13 AM
struck
Thanks a lot for that.. but I am not able to graph it :( ... (Headbang)

• Feb 13th 2008, 11:41 AM
Soroban
Hello, struck!

We assume: .$\displaystyle \begin{array}{cccc}x & \geq & 0 & {\color{blue}[1]} \\ y & \geq & 0 & {\color{blue}[2]} \end{array}$

We have: .$\displaystyle \begin{array}{cccccccc}25x + 15y & \leq & 1800 & \;\Rightarrow\; & 5x + 3y & \leq & 360 & {\color{blue}[3]}\\ x - 3y & \geq & 0 & \Rightarrow & y & \leq & \frac{1}{3}x & {\color{blue}[4]}\end{array}$

[1] and [2] places us in Quadrant 1.

Graph the line of [3]: .$\displaystyle 5x + 3y \:=\:360$
It has intercepts (72,0) and (0,120).
Graph the line and shade the region below it.

Graph the line of [4]: .$\displaystyle y \:=\:\frac{1}{3}x$
It contains the origin and has slope 1/3.
Graph the line and shade the region below it.

Your graph should look like this:
Code:

        |  (0,120)*         | *         |  *         |    *         |      *          *         |        *    *         |          o         |      *:::::*         |  *:::::::::::*       - o - - - - - - - - o - -         |              (72,0)

Can you finish it now?