Linear Programming: What system of equations allows the 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?

1 Answer
Nov 14, 2017

See below.

Explanation:

Calling

#S = 20# total area for planting

#c_A = 120# seed cost #A#
#c_B = 200# seed cost #B#

#x_A = # acres destined to crop #A#
#x_B = # acres destined to crop #B#

We have the restrictions

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

the total costs

#f_C = x_A c_A + x_B c_B + 15 xx 6.50 xx x_A + 10 xx 5.00 xx x_B#

and the expected income

#f_P =600 x_A + 200 x_B#

so the maximization problem can be stated as

Maximize

#f_P - f_C#

subjected to

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

and the solution gives #x_A = 15, x_B = 0# with a global profit of

#f_P-f_C = 5737.5#