Question

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?

Answer 1

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`