The Farming Project

If you have the choice between two different crops, how many acres of each do you farm?

Suppose you have inherited a plot of land between 100 and 10,000 acres.  Your job is to go through the following tasks to determine how you can maximize your profit.  In order to do so, you must consider the following:

• Your land has already been planted with well established pistachio trees.  The problem is that there is a limit as to how many acres of pistachios you can farm.  Because of certain restrictions, you will need to replace some of the pistachios with cotton. 
• How much does each crop cost to farm (per acre)?  And how much money is the bank willing to loan you per acre?
• How much money can be made per acre?  Keep in mind that one crop may demand a higher price, but may cost more to farm.
• How many units per acre of nitrogen will be needed to fertilize the crop? Is there any limit to the amount of nitrogen available to you?
• How much water does it take to grow each crop?
• How much water is available?  Remember, we live in the San Joaquin Valley and water doesn’t just fall from the sky…does it? J
• Are there any restrictions on how much of each crop a farmer can plant on one piece of land? 

Before you start planting, you must decide how much of each crop to farm. 

Let x = the number of acres devoted to cotton.

Let y = the number of acres of pistachios.  

How many acres of each do you want to farm in order to maximize your profit?

You should

Note: Have Mr. Cox check each task before moving on to the next task. 

Task #1: Search Google Earth for the perfect piece of land (Any shape other than a rectangle).  Once you have found it, take a snapshot of the land in Smart.  Determine the dimensions of the property that would give you your desired acreage.  Remember, you must be between 100 and 10,000 acres (round to the nearest 100 acres).  Determine the equations that would model the property lines.  You may use GeoGebra to help you with this, but you should also demonstrate how you would find those equations algebraically.  Equations must be in Standard and Slope Intercept Form.  (Standard 6.0 and 7.0)

Task #2:  The bank is willing to loan you $2000 per acre to farm your land.  However, cotton costs $1000 per acre and pistachios cost $3000 per acre to farm.  Determine how much money you have available for this project.  Note:  You must first determine how many acres you have to farm. How big is an acre?  Look it up! 

What inequality can be used to model this situation?  (Standard 2.0, 7.0)

Task #3:  Because of the high demand on fertilizer and water, you have a limit as to how much of each you can obtain.  Your fertilizer supplier can provide you with 340 units of fertilizer per acre and the water district will allot you 1.7 acre-feet of water per acre. Cotton requires 300 units of fertilizer per acre and 2 acre-feet of water per acre.  Because the pistachios are well established, they will require more fertilizer but less water.  Pistachios take 400 units of fertilizer per acre and 1 acre-foot of water per acre. 

Write two inequalities for this situation.  Let the first inequality represent the amount of nitrogen needed compared to the amount available.  Let the second inequality compare the amount of water needed with the amount available.  (Standard 7.0)

Task #4:   Because of the fact that you will be “changing” an existing piece of land, you will be required to adhere to a new state law that states that pistachios cannot take up more than 60%  and cotton cannot take up more than 80% of your land. 

Write a set of inequalities that model this.  Now you are ready to graph your set of linear inequalities.  But, before you do, there is one last inequality that you must consider.  Is ther a limit to how large x + y can be?GeoGebra doesn't handle inequalities very well so you must turn them into equations in order to graph.  Insert your equations into GeoGebra and use what you know about inequalities to determine the shaded region.  Use the polygon tool to create the polygon that is determined by the shaded region. (Standard 6.0)

Task #5:  How much money can you make?  The current selling prices for your crops are as follows:

            Cotton: $1500 per acre

            Pistachios: $4000 per acre

Write an equation that involves x and y that could be used to determine potential profit. 

Task #6:   The vertices of your polygon from Task #4 can be used to determine your maximum profit.  Use GeoGebra to determine the vertices of your polygon.  Once you have found the coordinates of each vertex, substitute the values of x and y into your profit equation to determine potential profits.  Which point gives you the most profit?  Which lines are used to determine this point?  Show how you could have found that point of intersection algebraically. (Standard 4.0 and 9.0)

Task #7:  In order to maintain your crop, you must spray an herbicide to control the weeds.  Glyphosate is a common herbicide used in agriculture.  However, glyphosate can be purchased in different concentrations.  A farmer can purchase a solution that is 54% glyphosate but your average homeowner can only purchase solution that is 12% glyphosate.  You happen to have thousands of gallons of both available but, new legislation dictates that you can only use a solution of 36% glyphosate. 

Your job is to determine how many gallons of 54% glyphosate must be mixed with 12% glyphosate in order to obtain a mixture that is 36% glyphosate.  The number of gallons of 36% glyphosate is dependent upon the number of acres you will be farming.  Keep in mind that you will only be spraying the land that is being farmed and you will use .38 gallons/acre.  (Standard 15.0)

Task #8:It is time to start pruning the trees and you hire three new workers.  James can prune a tree in 5 minutes, Jose can prune the tree in 3 minutes and Mark can prune a tree in 2 minutes.  If you have 136 trees per acre, how long will it take them to prune all the trees? Does this seem reasonable?  Why? How many 3 man crews (working at the same rate) would you need to hire in order to get the work done in four 54-hour work weeks? (Standard 15.0)

Task #9:  Create a final proposal justifying how many acres of each crop you will farm.  Your proposal should include but is not limited to the following: 

• Picture from Google Earth (you imported to GeoGebra) of the land you are purchasing with the lines and equations that determine the borders.
• Budget, Fertilizer, Water, State Law restrictions inequalities.
• Profit equation.
• Picture of your polygon from GeoGebra.  Include labels for the points and the equations you use.
• Written recommendation explaining your plan of action.  Be sure to give brief explanations behind your conclusions.  Your explanations do not have to be long, but they do need to justify your conclusions. 
• Your final proposal must be digital and able to be embedded into a webpage. You may use Voicethread, Slideshare, Screencast, Prezi or any other tool agreed upon between you and Mr. Cox. 

Farming With Google by David Cox is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

Based on a work at coxmath.pbwiki.com.