document.write( "Question 971810: Use the graphical method to solve the following linear programming problem. Restrict x ≥ 0 and y ≥ 0.
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\n" ); document.write( "Maximize
\n" ); document.write( "f = 30x + 27y
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\n" ); document.write( "30x + 10y ≤ 180
\n" ); document.write( "20x + 20y ≤ 200
\n" ); document.write( "x≥0, y≥0
\n" ); document.write( "a.) Graph the solution of the constraint system
\n" ); document.write( "b.) Find the corners of the resulting feasible region
\n" ); document.write( "c.) Evaluate the objective function at each corner
\n" ); document.write( "d.) Indicate the corner which provides the Maximum value for the objective function \n" ); document.write( "
Algebra.Com's Answer #594339 by Edwin McCravy(20056)\"\" \"About 
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Maximize
\n" ); document.write( "f = 30x + 27y
\n" ); document.write( "Subject To:
\n" ); document.write( "30x + 10y ≤ 180
\n" ); document.write( "20x + 20y ≤ 200
\n" ); document.write( "x≥0, y≥0
\n" ); document.write( "a.) Graph the solution of the constraint system
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document.write( "The boundary lines' equations:\r\n" );
document.write( "30x + 10y = 180 \r\n" );
document.write( "20x + 20y = 200\r\n" );
document.write( "        x = 0  (the equation of the y-axis)\r\n" );
document.write( "        y = 0  (the equation of the x-axis)\r\n" );
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document.write( "These simplify to\r\n" );
document.write( "3x + y = 18 which has intercepts (0,18) and (6,0)\r\n" );
document.write( " x + y = 10 which has intercepts (0,10) and (10,0)\r\n" );
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document.write( "We only need to draw the 1st quadrant since neither variable\r\n" );
document.write( "is negative, which is what x≥0, y≥0 tells us.\r\n" );
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document.write( "We will also find the point where the slanted lines intersect\r\n" );
document.write( "by solving this system:\r\n" );
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document.write( "\"system%283x%2By=18%2Cx%2By=10%29\"\r\n" );
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document.write( "Use substitution or elimination to get x=4, y=6, so they intersect \r\n" );
document.write( "at the point (x,y) = (4,6)\r\n" );
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\n" ); document.write( "b.) Find the corners of the resulting feasible region
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document.write( "Since the first two inequalities have ≤, the feasible region is the region\r\n" );
document.write( "below the two slanted lines, to the right of the y-axis and above the x-axis.\r\n" );
document.write( "Let's eliminate all of the graph except for the feasible region:\r\n" );
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\n" ); document.write( "c.) Evaluate the objective function at each corner
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document.write( "Corner |  \r\n" );
document.write( "point  |f(x) = 30(x) + 27(y)  =    VALUE      \r\n" );
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document.write( "(0,0)  |       30(0) + 27(0)  =   0 +   0 =   0   <---minimum value = 0\r\n" );
document.write( "(0,10) |       30(0) + 27(10) =   0 + 270 = 270\r\n" );
document.write( "(4,6)  |       30(4) + 27(6)  = 120 + 162 = 282   <---maximum value = 282\r\n" );
document.write( "(6,0)  |       30(6) + 27(0)  = 180 +   0 = 180\r\n" );
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\n" ); document.write( "d.) Indicate the corner which provides the Maximum value for the objective function.
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document.write( "The maximum value is 282 which is reached at the corner point (4,6).\r\n" );
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document.write( "Edwin
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