This is a word problem that I need help solving

Question

Juan makes two types of wood clocks to sell at local stores. It takes him 2 hours to assemble a pine clock, which requires 1 ounce of varnish. It also takes 2 hours to assemble an oak clock, which takes 4 ounces of varnish. Juan has 16 ounces of varnish in stock, and he can work 20 hours. If he makes $3 profit on each pine clock and $4 on each oak clock, how many of each type should he make to maximize his profit?

Answer 1

This is a linear programming problem.

 

Let p = the number of pine clocks and k = the number of oak clocks.

 

The total profit is the "objective function" -- the thing you want to maximize.  That's equal to 3p + 4k.

 

What else do you know?

 

Since a pine clock uses 1 ounce of varnish and an oak clock uses 4, the total amount of varnish he uses is going to be p + 4k and that can't be more than 16, since that's all the varnish he has.  So one constraint is p + 4k ≤ 16.

 

Since each kind of clock requires 2 hours, the total amount of time he takes is going to be 2p + 2k and that can't be more than 20, since that's the maximum amount of time he can work.  So another constraint is 2p + 2k ≤ 20.  You can divide through by 2 to get a simpler inequality:  p + k ≤ 10.

 

You also know that the number of each kind of clock can't be negative, since he can't make a negative number of anything.  So that results in two more constraints:  p ≥ 0 and k ≥ 0.

 

So the problem is:

 

Maximize 3p + 4k subject to the following constraints:

 

p + 4k ≤ 16

p + k ≤ 10

p ≥ 0

k ≥ 0

 

There are a variety of ways to solve this.  One of the ways that helps you visualize it the best is to graph it.

 

So first graph all four of the inequalities on graph paper.  That will give you a region surrounded by four intersecting line segments (called the "feasible region"), so it will be a 4-sided region.  There will be 4 vertices, points where two lines intersect.  The solution to a linear programming problem has to be at one of the vertices.

 

Now figure out the coordinates of each vertex.  That will give you an x and y value for each of the four points.  Substitute each x-y pair into the objective function expression and see which vertex gives you the maximum value of the objective function.  Whichever one that is, that is your answer.