# 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!

Nov 15, 2017

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