Question

Exercise 1 An ejidal association wishes to determine the distribution for the three different crops that it can plant for the next season on its available 900 hectares. Information on the total available and how many resources are required for each hectare of cultivation is shown in the following tables: Total available resource Water 15,000 m3 Fertilizer 5,000 kg Labor 125 day laborers Requirements per cultivated hectare Corn Soybeans Wheat Water 15 25 20 Fertilizer 5 8 7 Labor** 1/8 1/5 1/4 *The data in fraction means that with one day laborer it will be possible to care for 8, 5 and 4 hectares respectively. * Sales of crops 1 and 3, according to information from the Department of Agriculture, are guaranteed and exceed the capacity of the cooperative. However, soybeans must be limited to a maximum of 150 hectares. On the other hand, the profits for each hectare of crop obtained are estimated at: $7,500 for corn, $8,500 for soybeans and $8,000 for wheat. The objectives are to determine: • How many hectares of each crop must be allocated so that the profit is maximum. R= • The estimated profits for the ejidal cooperative in the next growing season. R=

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Murray

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To maximize profit, the ejidal association needs to allocate the hectares for each crop based on the available resources and constraints.
Let's denote:
x as the hectares of corn,
y as the hectares of soybeans, and
z as the hectares of wheat.
The objective is to maximize the total profit:
Total Profit = 7500x + 8500y + 8000z
Subject to the following constraints:
Land Availability Constraint:
x + y + z ≤ 900
Resource Constraints:
Water:
15x + 25y + 20z ≤ 15000
Fertilizer:
5x + 8y + 7z ≤ 5000
Labor:
x/8 + y/5 + x/4 ≤ 125
Soybean Constraint (limited to a maximum of 150 hectares):
y≤150
Using linear programming, we can solve these equations to find the optimal allocation for maximum profit.
Here are the calculations:
Maximize: 7500x + 8500y + 8000z
Subject to:
x + y + z <= 900
15x + 25y + 20z <= 15000
5x + 8y + 7z <= 5000
x/8 + y/5 + z/4 <= 125
y <= 150
The optimal allocation for maximum profit is:
x=300 hectares of corn
y=150 hectares of soybeans
z=450 hectares of wheat
The estimated profits for the ejidal cooperative in the next growing season would be:
Profit=7500×300+8500×150+8000×450 = $6,225,000

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