SOLUTION: Find the values of x and y that minimize the objective function C=2x + 3y for these constraints y>=x x+y<=32 x>= 5, y >= 3 A. C = 91 for (5,27) B. C = 80 for (16,16) C

Question 1061813: Find the values of x and y that minimize the objective function
C=2x + 3y for these constraints
y>=x
x+y<=32
x>= 5, y >= 3
A. C = 91 for (5,27)
B. C = 80 for (16,16)
C. C = 26 for (6,8)
D. C = 25 for (5,5)
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
THE QUICK SOLUTION:
You check the choices
Point (5,5), , yields .
That corresponds to choice ,
which has the least of the values listed for the objective function .

IF YOU HAVE TO SHOW YOUR WORK:
You can start by graphing the equations ,
, , , and ,
that represent the boundary lines (and part of the solutions) for the constraint inequalities.
The graph would look like this;
Then you figure out what "feasible region" is defined by the constraints.
I expect that region to be bounded on all sides,
so that would be the triangle CDE, bordered by three of the lines,
or the quadrilateral bordered ABDE, bordered by all four lines.
One way to determine the feasible region would require looking at each inequality separately,
and deciding what side of the corresponding boundary line is the solution.
For , and , it is obvious that the solutions are to the right, and above the line respectively.
For , we know that the solution includes, the origin, ,
because its coordinates satisfy : .
For , the solution is the space above the line .
So, the feasible region is triangle DCE.
At this point, you are expected to calculate the value of at the vertices of CDE.
You find the coordinates of the vertices by solving easy systems of linear equations.
Point D at the intersection of and has coordinates ,
solution to the system .
Similarly, you solve to find the coordinates of C,
and to find the coordinates of E.
You calculate for the coordinates of each of those vertices.
Which vertex gives you the minimum value for ?
In this case, , yielding , is the vertex that gives you the least value for .
Point D wins.
That vertex and that value is the solution.
If two of those vertices were tied for the win,
the minimum would be the entire line segment connecting them.

NOTE:
Each value of the objective function happens along a line described by
.
Those lines are parallel to each other, with slope ,
sloping down like (which has a slope of ),
but just a little less steep.
The lower the value of , the lower the line.
In this case, that would tell you the answer is point D,
but your teacher would want you to do the calculations for points C and E, anyway.
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