# 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?

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

${f}_{C} = {x}_{A} {c}_{A} + {x}_{B} {c}_{B} + 15 \times 6.50 \times {x}_{A} + 10 \times 5.00 \times {x}_{B}$

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

${f}_{P} - {f}_{C} = 5737.5$