# system of inequalities

• Apr 19th 2009, 08:16 PM
lightninghelix
system of inequalities
i have this problem

A stationary company makes two types of notebooks: a deluxe notebook with subject dividers, which sells for \$4.00, and a regular notebook, which sells for \$3.00. The production cost is \$3.20 for each deluxe notebook, and \$2.60 for each regular notebook. The company has the facilities to manufacture between 2000 and 3000 deluxe notebooks and between 3000 and 6000 regular notebooks, but not more than 7000 altogether. How many notebooks of each type should be manufactured to maximize the difference between the selling prices and the production costs.

can someone help me get the system of inequalities i need to use to find the max.
thanks.
• Apr 20th 2009, 12:28 AM
Hello lightninghelix

Suppose they make $x$ deluxe notebooks and $y$ regular notebooks.
Quote:

a deluxe notebook with subject dividers, which sells for \$4.00, and a regular notebook, which sells for \$3.00.
The total selling price is \$ $(4x+3y)$.
Quote:

The production cost is \$3.20 for each deluxe notebook, and \$2.60 for each regular notebook
So the total production cost is \$ $(3.2x+2.6y)$
Quote:

between 2000 and 3000 deluxe notebooks
$2000 \le x \le 3000$
Quote:

between 3000 and 6000 regular notebooks
$3000\le y \le 6000$
Quote:

not more than 7000 altogether
$x+y\le 7000$

Can you continue from here?

• Apr 20th 2009, 07:26 AM
lightninghelix
so i would graph the lines 4x+3y less than or equal to 7000 and 3.2x+2.6y less than or equal to 7000 ?

wait i forgot about the objective functions, so i think i would graph all the inequalities you gave then i would use the vertices in those and plug them into both objective functions? which are the production and selling costs?
• Apr 20th 2009, 10:45 AM
Inequalities
Hello lightninghelix
Quote:

Originally Posted by lightninghelix
so i would graph the lines 4x+3y less than or equal to 7000 and 3.2x+2.6y less than or equal to 7000 ?

wait i forgot about the objective functions, so i think i would graph all the inequalities you gave then i would use the vertices in those and plug them into both objective functions? which are the production and selling costs?

Yes. Plot the inequalities and work out the region in which the possible values of $x$ and $y$ lie. (It's a trapezium shape.) Then for the objective function, you need the profit, i.e. difference between the production costs and the selling price; that's

$P = (4x+3y) - (3.2x + 2.6y) = 0.8x + 0.4y$

and find the values of $x$ and $y$ in the region that makes $P$ a maximum. (I think the answer is $x = 3000$ and $y = 4000$.)