Thread: Linear Programming and systems, HELP ME! :(

This problem is so hard. I can't figure it out!
"A gaming manufacturing company has developed a new gaming system. To produce the new system, they plan on using resources in two manufacturing plants. The table gives the hours needed for three tasks. For both plants combined, the company has allocated the following resources on a weekly basis: 1700 h of motherboard production, 1800h of technical labor, and 2400 h of general manufacturing. The first plant earns a profit of $90 per gaming system and the second plant earns $70 per system.

Resources	Plant 1 (Hours per system)	Plant 2 (hours per system)
Motherboard Production	9	1
Technical Labor	9	3
General manufacturing	4	8

Use the information above to determine how many gaming systems the company should make in each plant to maximize profit.
1. Create an objective function for the profit P that the company can earn. Let x represent the number of gaming systems that will be made in Plant 1, and let y represent the number of gaming systems that will be made in plant 2.
(My answer - P(x,y)=90x +70y

2. Write a constraint function for each of the resources and for any contextual contraints that you identify.
my answers:
9x+1y<_ 1700
9x+3y<_ 1800
4x+8y<_ 2400

3.Graph the constraint functions. Then use systems of equations to find the vertex points of the feasibility region.

4. Which vertex point maximizes profit with the given constraints.

5. What is the maximum profit that the company can make with the given constraints? How many gaming systems should each plant make to maximize profit?


I need help with numbers 3, 4, and 5!! :/

Hey jessicahatesmath.

For 3) you will have three sets of inequalities and all you have to do is find the area that is common to all inequalities.

So in these examples, the easiest way to find the inequality is find the equality and then see which half is > and which half is <.

So in the equality we have 9x + 1y = 1700. To find this line we get two points on the line and the line is a straight line going through those two points.

So x = 0 means y = 1700 so one point is (0,1700). When y = 0 we get 9x = 1700 so x = 188.8889 (roughly) so another point is (1888.8889,0)

Now you can graph this line by joining a straight line between the two points using a ruler (if you don't have a computer program to do it).

Now you want 9x + y <= 1700 so we need to see whether above the line or below the line satisfies the inequality. Lets pick the origin (0,0). Then 9*0 + 1*0 = 0 <= 1700 so the region for this inequality is below the line formed by the two points (0,1700) and (188.8889).

Now do this for the other lines and see what region they all have in common.

Thanks, i figured out how to do it. My coordinates are
(0,1700), (0,188)
(0,600), (0.3, 200)
(0,300), (0, 600)

Im not sure if thats the answer. But now im stuck on problem 4. Which Vertex point maximizes profit with the given constraints? Will the answer be the coordinate where they all intersect?

Just plug in the values for the inputs corresponding to that vertex to get a value for the profit and compare the differences between the vertices.