document.write( "Question 665012: A factory can produce two products, x and y, with a profit approximated by P = 14x + 22y – 900. The production of y can exceed x by no more than 200 units. Moreover, production levels are limited by the formula x + 2y ≤ 1600. What production levels yield maximum profit? \r
\n" ); document.write( "\n" ); document.write( "A. x = 400; y = 600
\n" ); document.write( "B. x = 0; y = 0
\n" ); document.write( "C. x = 1,600; y = 0
\n" ); document.write( "D. x = 0; y = 200 \n" ); document.write( "

Algebra.Com's Answer #413706 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
Obvious constraints are:
\n" ); document.write( " $\"x%3E=0\"$ and $\"y%3E=0\"$
\n" ); document.write( "(no negative production)
\n" ); document.write( "Given constraints are:
\n" ); document.write( " $\"y-x%3C=200\"$
\n" ); document.write( "( $\"y\"$ cannot exceed $\"x\"$ by more than $\"200\"$ ,
\n" ); document.write( "the difference has to be $\"200\"$ or less),
\n" ); document.write( "and
\n" ); document.write( " $\"x%2B2y%3C=1600\"$ .
\n" ); document.write( "
\n" ); document.write( "Because of those constraints,
\n" ); document.write( "there is a feasibility region.
\n" ); document.write( "You only can work in that region of the x-y plane.
\n" ); document.write( "That region is bordered by the lines represented by
\n" ); document.write( " $\"x=0\"$ , $\"y=0\"$ , $\"y-x=200\"$ , and $\"x%2B2y=1600\"$ .
\n" ); document.write( "You can graph the lines.
\n" ); document.write( " $\"graph%28300%2C300%2C-200%2C1800%2C-100%2C900%2C200%2Bx%2C800-0.5x%29\"$
\n" ); document.write( " $\"y-x=200\"$ graphs as the red slanted line ,
\n" ); document.write( "and $\"x%2B2y=1600\"$ graphs as the green line.
\n" ); document.write( "Your feasibility region is tue quadrilateral with
\n" ); document.write( "parts of the x-axis, the y-axis and the green and red lines for sides.
\n" ); document.write( "You can find the intersection points for each pair of lines.
\n" ); document.write( "For example, solving $\"system%28y-x=200%2Cx%2B2y=1600%29\"$
\n" ); document.write( "gives you the solution $\"x=400\"$ with $\"y=600\"$
\n" ); document.write( "for point (400,600), where the red and green slanted lines intersect.
\n" ); document.write( "
\n" ); document.write( "The vertices of your feasibility region are:
\n" ); document.write( "(0,0) , (0,200) , (400,600) and (1600,0).
\n" ); document.write( "The maximum for $\"P\"$ will happen at 1 of those points.
\n" ); document.write( "(In some cases it could happen at 2 of vertices and the whole segment connecting them).
\n" ); document.write( "All you need to do is calculate $\"P\"$ for each of those 4 points.
\n" ); document.write( "I will show you the calculation for 2 of them:
\n" ); document.write( "For point (400,600), with $\"x=400\"$ and $\"y=600\"$ ,
\n" ); document.write( " $\"P=14%2A400%2B22%2A600-900=17900\"$ .
\n" ); document.write( "For point (1600,0), $\"P=14%2A1600%2B22%2A0-900=21500\"$ .
\n" ); document.write( "The other points give you smaller values for $\"p\"$ ,
\n" ); document.write( "so the solution is point (1600,0),
\n" ); document.write( "with $\"highlight%28x=1600%29\"$ and $\"highlight%28y=0%29\"$ .
\n" ); document.write( "
\n" ); document.write( "THE REASON WHY IT WORKS THAT WAY:
\n" ); document.write( "The function to maximize, $\"P=14x%2B22y-200\"$
\n" ); document.write( "is a function of x and y,
\n" ); document.write( "which could be represented in 3 dimensions,
\n" ); document.write( "wit $\"z=14x%2B22y-200\"$ being the third, dependent variable.
\n" ); document.write( "As with altitude as a function of 2-dimensional coordinates (latitude and longitude),
\n" ); document.write( "we can represent the function $\"P%28x%2Cy%29=14x%2B22y-900\"$ on paper by making a contour map.
\n" ); document.write( "The contour lines would be $\"14x%2B22y-200=K+for+various+values+of+the+constant+%7B%7B%7BK\"$ .
\n" ); document.write( "Luckily for us, the function is linear in $\"x\"$ and $\"y\"$ ,
\n" ); document.write( "so those contour lines will be straight lines, like the blue line below.
\n" ); document.write( " $\"graph%28300%2C300%2C-200%2C1800%2C-100%2C900%2C200%2Bx%2C800-0.5x%2C%2810000-14x%29%2F22%29\"$
\n" ); document.write( "The blue line is the graph of
\n" ); document.write( " $\"14x%2B22y=10000\"$ <---> $\"14x%2B22y-900=9100\"$
\n" ); document.write( "That is the line for $\"P=9100\"$ .
\n" ); document.write( "As you change the constant, the line changes,
\n" ); document.write( "but all the other contour lines are parallel to that blue line.
\n" ); document.write( "As increase the value for the constant the lines moves away from the origin, until it moves out of your feasibility region.
\n" ); document.write( "You want the values (for $\"x\"$ , $\"y\"$ and $\"P\"$ for the largest possible $\"P\"$ ,
\n" ); document.write( "when you reach the end of the feasibility region.
\n" ); document.write( "In general, that will happen at one of the vertices, or at 2 of the vertices and the side that joins them.
\n" ); document.write( "In this case it happens at point (1600,0).
\n" ); document.write( "The maximum for P will be found at (1600,0)
\n" ); document.write( "where $\"P=14%2A1600-22%2A0-900=21500\"$ \r
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