document.write( "Question 203030: A fruit grower has 150 acres of land available to raise two crops, A and B. It takes 1
\n" ); document.write( "day to trim an acre of crop A and 2 days to trim an acre of crop B, and there are 240 days available each
\n" ); document.write( "year for trimming. It takes 0.3 days to pick an acre of crop A and 0.1 days to pick an acre of crop B, and
\n" ); document.write( "there are 30 days per year available for picking. The profit is $140 per acre for crop A and $235 per acre
\n" ); document.write( "for crop B. Determine the optimal acreage for each fruit and the optimal profit. \n" ); document.write( "
Algebra.Com's Answer #153175 by Theo(13342)\"\" \"About 
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let x = number of acres of crop A.
\n" ); document.write( "let y = number of acres of crop B.
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\n" ); document.write( "the equation we want to maximize is the profit equation.
\n" ); document.write( "the profit on crop A is 140 per acre so the total profit on crop A would be:
\n" ); document.write( "140 * x
\n" ); document.write( "the profit on crop B is 235 per acre so the total profit on crop B would be:
\n" ); document.write( "235 * y
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\n" ); document.write( "it takes 1 day to trim an acre of crop A and 2 days to trim an acre of crop B with a total of 240 days available for trimming.
\n" ); document.write( "1*x + 2*y <= 240
\n" ); document.write( "solving for y, we get:
\n" ); document.write( "y <= (240-x)/2
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\n" ); document.write( "it takes .3 days to pick an acre of crop A and .1 days to pick an acre of crop B with a total of 30 days available for picking.
\n" ); document.write( ".3*x + .1*y <= 30
\n" ); document.write( "solving for y we get:
\n" ); document.write( "y <= 300 - 3*x
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\n" ); document.write( "total number of acres has to be less than or equal to 150.
\n" ); document.write( "x + y <= 150
\n" ); document.write( "solving for y, we get
\n" ); document.write( "y <= 150
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\n" ); document.write( "the 3 constraints are shown in the following graph.
\n" ); document.write( "one of the points of intersection of all 3 lines should contain the maximum profit.
\n" ); document.write( "-----
\n" ); document.write( "since the points of intersection cannot be clearly seen, they can be calculated by taking sets of 2 equations and solving them simultaneously.
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\n" ); document.write( "equation 1 is: 1x + 2y <= 240_____ solving for y gets y <= (240-x)/2
\n" ); document.write( "equation 2 is: .3x + .1y <= 30____ solving for y gets y <= 300 - 3x
\n" ); document.write( "equation 3 is: x + y <= 150_______ solving for y gets y <= 150 - x
\n" ); document.write( "-----
\n" ); document.write( "for equation 1 and 2, the point of intersection = (72,84)
\n" ); document.write( "for equation 1 and 3, the point of ijntersection = (60,90)
\n" ); document.write( "for equation 2 and 3, the point of intersection = (75,75)
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\n" ); document.write( "\"graph+%28600%2C600%2C-30%2C120%2C-30%2C120%2C150-x%2C300-3x%2C%28240-x%29%2F2%29\"\r
\n" ); document.write( "\n" ); document.write( "since the maximum profit should lie at the intersection of these 3 lines, we can solve the profit equation at each of these points and pick the one that is maximum.
\n" ); document.write( "-----
\n" ); document.write( "the profit equation is again:
\n" ); document.write( "140x + 235y
\n" ); document.write( "----
\n" ); document.write( "at point (72,84) the profit would be:
\n" ); document.write( "140*72 + 235*84 = $29,820
\n" ); document.write( "at point (60,90) the profit would be:
\n" ); document.write( "140*60 + 235*90 = $29,550
\n" ); document.write( "at point (75,75) the profit would be:
\n" ); document.write( "140*75 + 235*75 = $28,125
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\n" ); document.write( "It appears that maximum profit would be attained with 72 acres of crop A and 84 acres of crop B but this can't be because 72 + 84 > 150 which violates the maximum number of acres constraint (equation 3).
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\n" ); document.write( "The next choice would be 60 acres of crop A and 90 acres of crop B.
\n" ); document.write( "60 + 90 = 150 <= 150 so the constraint on maximum number of acres is not violated (equation 3).
\n" ); document.write( "60 + 2*90 = 60 + 180 = 240 <= 240 so the constraint on maximum number of days to trim is not violated (equation 1)
\n" ); document.write( ".3*60 + .1*90 = 18 + 9 = 27 <= 30 so the constraint on maximum number of days to pick is not violated (equation 2)
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\n" ); document.write( "It appears the answer is:
\n" ); document.write( "60 acres of crop A and 90 acres of crop B gives the maxiumum profit while meeting all the constraints.
\n" ); document.write( "the maximum profit is $29,550
\n" ); document.write( "-----
\n" ); document.write( "It appears to be logically correct since the profit on crop B acres is much greater than the profit on crop A acres so maximizing the acres for crop B makes sense.
\n" ); document.write( "-----
\n" ); document.write( "based on the constraints it appears the maximum acres of crop B are 90. I tried 91 acres of crop B and came up with a smaller profit because of lower total utilization on the acres that could be cropped (91 for B left only 58 for A because of the constraints). I suspect further analysis would yield the same results, i.e. the max profit is what is shown.
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