### Optimization of Three Crops

## by Alyssa Domek

Submitted : Fall 2011

This project was proposed to help a farmer fully optimize all three of his crops to gain the most profit without exceeding his 23 acre limit. For this project to be successfully addressed I took note of all the given information from the problem and made sure to record every formula correctly. From there I plugged in the given information into the main formula of Profit= C_{A}A_{A} + C_{B}A_{B} + C_{C}A_{C} + β_{AB}A_{A}A_{B }+ β_{AC}A_{A}A_{C }+ β_{BC}A_{B}A_{C }where I was left with two unknowns. After plugging in other given information that will be explained soon, I then took the partial derivative with respect to A_{B} first to acquire how many acres of Crop B would be allowed.

After I found A_{B} I then took the partial derivative with respect to A_{C} from which I was able to obtain the total acres of Crop C that would be allowed. From there I proceeded to insert my answers into the equation A_{A}+A_{B}+A_{C}=23 from which I was able to attain A_{A} and establish the number of acres of Crop A the farmer would be allowed to plant to achieve the maximum profit. To check my work I proved that I calculated the maximum acreage by solving the determinant matrix and I concluded that I was correct.

[ Back ]