Question 1155591
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Let x = number of soldiers
Let y = number of guns<br>
The constraints are....
(1) 10x+4y <= 60  (maximum units of wood)
(2) 22 <= 4x+6y <= 44  {minimum and maximum units of plastic)
(3) x <= 2  (maximum demand for soldiers)
(4) y >= 2x  (at least 2 guns for each soldier)<br>
and, of course, x>=0 and y>=0<br>
Sketch a graph of the constraint boundary lines and the resulting feasibility region.<br>
Contrary to what is usually taught, it is NOT necessary to evaluate the objective function at every corner of the feasibility region.<br>
Instead, the corner at which the objective function is maximized can be determined by comparing the slopes of the constraint boundary lines and the objective function.<br>
The objective function is P = 30x+20y; its slope is -3/2.  The maximum value of the objective function will be obtained at the corner of the feasibility region where a line with slope -3/2 just touches the feasibility region.<br>
If you have drawn your graph carefully and know the slopes of the constraint boundary lines, you will see that happens at the intersection of the constraint boundary lines x=2 and 4x+6y=44.  The coordinates of that corner are (2,6).<br>
The value of the objective function at that corner is 30(2)+20(6) = 60+120 = 180.<br>
ANSWERS:  The maximum profit is $180, when 2 soldiers and 6 guns are produced.<br>