Linear Programming: What acreage allows to farmer to maximize profit?

A farmer has a choice of planting a combination of two different crops on 20 acres of land. For crop A, seed costs $120 per acre, and for crop B, seed costs $200 per acre. Government restrictions limit acreage of crop A to 15 acres but do not limit crop B. Crop A will take 15 hours of labor per acre at a cost of $5.60 per hour, and crop B will require 10 hours of labor per acre at $5.00 per hour. The expected income from crop A is $600 per acre, and crop B is $250 per acre. How many acres of each crop should the farmer plant in order to get maximum profit?
I know the answer is 15 acres of A and 5 acres of B, but I don't know how to get there...

1 Answer
Nov 15, 2017

See below.


Ignoring the costs and considering only the profits you can equate

#max 600 x_A + 250 x_B#

subjected to

#x_A ge 0#
#x_B ge 0#
#x_A le 15#
#x_A+x_B le 20#


#x_A = # planted acres of crop #A#
#x_B = # planted acres of crop #B#

giving as optimum result

#x_A=15, x_B = 5#

Attached a plot

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