Question

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...
Thanks!

Answer 1

See below.

Explanation:

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`

where

`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