Problema Solution
A farmer has 100 acres, he will plant corn and wheat. He must plant at least 20 acres of corn. He has 3600 pounds of fertilizer. The corn needs 30 pounds of fertilizer per acre; wheat needs 40 pounds. His profits ate $150/acre of corn and $200/acre of wheat. How much should he plant of each crop to maximize his profit.
Answer provided by our tutors
let
c = acres of corn, c >= 0
w = acres of wheat, w >= 0
a farmer has 100 acres, he will plant corn and wheat
c + w <= 100
he must plant at least 20 acres of corn.
c >= 20
he has 3600 pounds of fertilizer. the corn needs 30 pounds of fertilizer per acre; wheat needs 40 pounds
30c + 40d <= 3600
the profit is determined by the objective function
F(c , w) = 150c + 200w
lets draw the graph of the system of inequalities
w >=0
c + w <= 100
c >= 20
30c + 40w <= 3600
click here to see the graph and find the corner points!
there are 4 corner points and we calculate F(c ,w) for each point to find the maximum value (the biggest value for F):
(20, 0) corner point
F(20, 0) = 150*20 + 200*0 = $3,000
(100, 0) corner point
F(100, 0) = 150*100 + 200*0 = $15,000
(40, 60) corner point
F(40, 60) = 150*40 + 200*60 = $18,000
(20, 75) corner point
F(20, 75) = 150*20 + 200*75 = $18,000
to maximize the profit the farmer should plant 40 acres of corn and 60 acre of wheat or 20 acres of corn and 75 acres of wheat.