Question 1182966
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The solution from the other tutor is fine.<br>
However, there is a refinement of the process he shows that cuts down a bit on the amount of work you need to do to finish the problem.<br>
Every resource you will find on linear programming says that, after finding the corners of the feasibility region, you have to evaluate the objective function at all the corners to find the maximum value of the objective function.<br>
That is not true.<br>
The corner where the objective function is maximized can be determined by comparing the slope of the objective function to the slopes of the constraint boundary lines.<br>
In this problem, we have, for the equations of the constraint boundary lines,
x+y = 4000 ==> y = -x+4000 ==> slope -1
2x+3y = 9000 ==> y = (-2/3)x+3000 ==> slope -2/3<br>
For the objective function, we have
25x+30y = C ==> y = (-5/6)x+C/30 ==> slope -5/6<br>
Since the slope of the objective function is between the slopes of the two constraint boundary lines, the maximum value of the objective function will be at the intersection of the two constraint boundary lines.<br>
It can be seen that this is the case by looking at the graph.  Of all the lines with slope -5/6, the one which "first touches" the feasibility region will be the one that touches the corner of the feasibility region that is the corner where the two lines that intersect at that corner have slopes for which the slopes of the two lines are one greater than and one less than -5/6.<br>
So to find the maximum value of the objective function, you only need to evaluate it at (3000,1000)<br>