Expert: Paul Klarreich - 2/21/2006

Question
A company manufactures checker sets and chess sets. Suppose each day the company has available 1900 boards (which can be used for both games) and 80,000 units of wood for making pieces. Each checker set uses 20 units of wood and each chess set uses 60 units of wood. The distributors the company sells to can take up to 1250 checker sets per day and up to 750 chess sets per day. The company makes a profit of $1.00 on each checker set and $2.00 on each chess set.

How many checker sets and how many chess sets should the company make each day in order to maximize its profits?

So far i have the following
x=number of checker sets
y= number of chess sets
x+y<=1900
20x+60y<=80000
P=1.00x +2.00y  

Answer
Hi, Kevin,
Your Question:  A company manufactures checker sets and chess sets. Suppose each day the company has available 1900 boards (which can be used for both games) and 80,000 units of wood for making pieces. Each checker set uses 20 units of wood and each chess set uses 60 units of wood. The distributors the company sells to can take up to 1250 checker sets per day and up to 750 chess sets per day. The company makes a profit of $1.00 on each checker set and $2.00 on each chess set.

How many checker sets and how many chess sets should the company make each day in order to maximize its profits?

So far i have the following
x=number of checker sets
y= number of chess sets
x+y<=1900
20x+60y<=80000
P=1.00x +2.00y
-------------------------------

That's better.  You have now discovered one of the major reasons for studying mathematics.  It forces you to pay attention to details -- an important life skill.

In your 'so far', you have not used all the facts in the problem.  It also says:

The distributors can take up to 1250 checker sets per day.

So  x <= 1250

The distributors can take up to 750 chess sets per day.

and  y <= 750

Now your collection of 'constraints' looks like this:

 x +  y <=  1900
20x +60y <= 80000  or:  x + 3y <= 4000
      x <= 1250
      y <= 750

So here's what you do:

A. Graph the four inequalities and note the 'feasible' region enclosed by all of them.

B. Write the associated equations (change <= to =) and solve each of the pairs of equations.  This will be some work and may force you to review solving simultaneous equations.

C. Consider the solution points.  Make sure each is in the feasible region and then,for each point, compute P = 1.00x + 2.00y

Whichever point gives the largest P is your solution.

Let me know if you get stuck anywhere.