Task 1
Get a plot of land, import it into GeoGebra, and demonstrate how to algebraically find the equation
of a few lines that make up its borders.
Task 2
Create an inequality that shows how to achieve maximum profit
Brief Explanation:
Since we have 200 acres the total amount that the bank will loan us is $400,000. The cotton costs $1000 per acre, and the pistachios cost $3000 per acre. The x and y represent how many acres of each crop we can plant with out exceeding the limit of 200 acres. 1000x + 3000y can't exceed $400,000.
Task 3
Part 1: Create an inequality that shows the amount of nitrogen needed compared to the amount available
Brief Explanation
Since we have 200 acres, the total amount of nitrogen we can obtain is 68,000 units per acre, where cotton (x) needs 300 units of nitrogen per acre, and pistachios (y) need 400 units of nitrogen per acre. The x and y represent how many acres of each crop we can plant without exceeding the limit of 200 acres.
Part 2: Create an inequality that shows the amont of water compared to the amount available
Brief Explanation
Since we have 200 acres, the total amount of water we can obtain is 240 acre-feet of water, where cotton (x) needs 2 acre-feet of water per acre, and pistachios (y) need 1 acre-feet of water per acre. The x and y represent how many acres of each crop we can plant without exceeding the limit of 200 acres.
Task 4
Part 1: Create a set of inequalities that shows that pistachois cannot take up more than 60% of our land, and shows that cotton cannot take up more than 80% of our land according to a new state law.
Brief Explanation
Since our total land is 200 acres, we multiplied 200 by .8 and got 160. Therefore, x must be less than or equal to 160, because we cannot exceed 160 acres of cotton. Also, to achieve maximum profit, we must have at least one acre of cotton.
Brief Explanation
Since our total land is 200 acres, we multiplied 200 by .6 (because only 60% of our land can be pistachios) and got 120. Therefore, y must be less than or equal to 120, because we cannot exceed 120 acres of pistachios. Also, to achieve maximum profit, we must have at least one acre of pistachio.
Brief Explanation
This graph shows our possible acres of each crop, with x being cotton and y being pistachios, to achieve profit. One of these points represents maximum profit. We are going to disregard the points H, F, and G because Point A is obviously going to give us more than H because they have the same y value but A has a bigger x value and point E is going to give us more than F because they have the same x value but F has a bigger y value.
Task 5
Write an equation that involves x and y that could be used to determine potential profit.
Profit
Brief Explanation
X is equal to cotton and Y is equal to pistachios. P is equal to profit. The reason the equation is 500x + 1000y = P is because we can get $1500 per acre of cotton, but it costs $1000 to farm it, and we can make $4000 per acre of pistachios, but it costs $3000 to farm it. We can make $500 per acre of cotton and $1000 per acre of pistachios.
Task 6
These are the points that could give us maximum profit.
Profit for point A: 500(40) + 1000(120) = 140,000
Profit for point B: 500(88) + 1000(104) = 148,000
Profit for point C: 500(120) + 1000(80) = 140,000
Profit for point D: 500(140) + 1000(60) = 130,000
Profit for point E: 500(160) + 1000(20) = 100,000
Brief Explantion
The lines A and B intersect to form the point B, which gives us maximum profit. The equation for line A is x + 3y = 400, and the equation for line B is 3x + 4y = 680. We found the intersection point of the equations as followed:
Task 7
Task 8
We should farm 88 acres of cotton and 104 acres of pistachios. This will allow us to achieve maximum profit on our 200 acre plot of land. We found this solution by using a set of inequalities (shown above) which represent the restrictions that we were given. Using algebra in a real life application we were able to find a solution to this situation, and make a total prophit of 148,000 dollars.