Question 1005202
you did good.


what's left is to graph the constraints and find the intersection points of the corners of the feasible region and then to evaluate the objective function at those corner points to find the one that contains the greatest profit.


your objective function should be:


profit = 50x + 52y


with an ordinary graphing software tool, or if you grpah manually, you would do the following:


take your constraint equations and solve for y.


2.5x + 3y <= 4000 becomes y <= (4000 - 2.5x)/3


2x + y <= 2500 becomes y <= (2500 - 2x)/1


.75x + `1.25y <= 1500 becomes y <= (1500 - .75x)/1.25


you would then graph the equations, not the inequalities.


specifically, you would graph:


y = (4000 - 2.5x)/3
y = (1500 - .75x)/1.25
y = (1500 - .75x)/1.25


you would then print that graph out and shade the regions that either satisfy the inequality, or do not satisfy the inequality.


the advantage to shading the regions that do not satisfy the inequality is that the region that does satisfy the inequality is the only region that's not shaded.


this works best with graphing software that does allow you to graph inequalities since the region that satisfies the inequalities stands out much better when it is the only region that is not shaded.


once you have shaded the region that satisfies the inequalities, or shaded the region that does not satisfy the inequalities, you then look for the corner points of the region that does satisfy the inequalities.


the corner points are the intersection of the lines that bound this region.


once you have found the intersection points of the lines that bound this region, you would then evaluate the objective function at each of those corner points and determine which one give you the maximum profit.


if you have graphing software that allows you to graph inequalities, your job is a little easier and much quicker.


the graphing software i used is at www.desmos.com/calculator.


you can click on it here:


<a href = "http://www.desmos.com/calculator" target = "_blank">link to desmos graphing calculator</a>


i graphed the regions that did not satisfy the inequality.


specifically, where the equation was y <= (4000 - 2.5x)/3, ...


i graphed y >= (4000 - 2.5x)/3.


i did the same with the other inequalities.


desmos also allows you to show the coordinates of the intersection points.


the result is what you see below:


<img src = "http://theo.x10hosting.com/2015/112101.jpg" alt="$$$" </>


there were two other constraints not shown above that needed to be considered.


they were x >= 0 and y >= 0


these were necessary because profit can't be negative.


naturally, i also graphed the reverse to shade the region that did not satisfy the inequality.


just to show you what i mean by it being easier to see the region that satisfies th einequalities when  it is not shaded, i changed the inequalities to shade the region that satisfied the inequalities and that is what you are seeing below.


<img src = "http://theo.x10hosting.com/2015/112103.jpg" alt="$$$" </>


as you can see, it's the same region, but it's easier to pick out when i shade the region that does not satisfy the inequality than when i shade the region that does satisfy the inequality.


if you are ahading manually, this doesn't matter so much because you can control the region that you're shading easier.


the graphing software is less discriminate because it will shade a region below a line that you would manually be able to see that you only wanted to shade the region between the lines.


at each corner point, you will then evaluate the objective function that says that profit = 50x + 52y


you will find that the maximum profit is generated when x = 571.4 and y = 857.1


you should also find that the constraints are satisfied at those points.


you do that by evaluating the constraint equations at those points.


assembly = 4000 <= 4000
paint = 2000 <= 2500
packaging = 1500 <= 1500
571.5 >= 0
857.1 >= 0


all the constraints were satisfied.


another interesting tidbit about desmos you might find useful.


desmos is capable of graphing 2.5 + 3y <= 4000 directly.


you don't really have to solve for y.


i did so because other graphing software requires you to solve for y.


if you are using desmos, you will very quickly become spoiled and realize that you can get away with things that other graphing software won't let you get away with.


hopefully you are now aware of the process.


possibly you have another tool at your disposal that should make your job of graphing easier.  


just be aware that desmos lets you get away with things that other graphing software doesn't.