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Jackson Farms has allotted a certain amount of land for cultivating soybeans, corn, and wheat. Cultivating 1 acre of soybeans requires 2 labor-hours, and cultivating 1 acre of corn or wheat requires 6 labor-hours. The cost of seeds for 1 acre of soybeans is $12, the cost for 1 acre of corn is $20, and the cost for 1 acre of wheat is $8. If all resources are to be used, how many acres of each crop should be cultivated if the following hold?
(a)
1000 acres of land are allotted, 3600 labor-hours are available, and $12,800 is available for seeds.
acres of soybeans acres of corn acres of wheat
(b)
1400 acres of land are allotted, 5600 labor-hours are available, and $18,800 is available for seeds.
acres of soybeans acres of corn acres of wheat

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Answer:

To determine the number of acres of each crop that should be cultivated based on the given constraints, we can set up a system of equations considering the land, labor-hours, and seed costs available for each scenario. Here's how you can approach it:

(a) For 1000 acres of land, 3600 labor-hours, and $12,800 for seeds:

Let x be the acres of soybeans, y be the acres of corn, and z be the acres of wheat.

1. The total land constraint:

x + y + z = 1000

2. The total labor-hour constraint:

2x + 6y + 6z = 3600

3. The total seed cost constraint:

12x + 20y + 8z = 12800

By solving this system of equations, you can determine the optimal number of acres of each crop to cultivate.

(b) For 1400 acres of land, 5600 labor-hours, and $18,800 for seeds:

Similarly, set up a system of equations with the corresponding constraints for this scenario and solve for the optimal number of acres of each crop to be cultivated.

By following these steps and solving the systems of equations, you can find the appropriate allocation of acres for soybeans, corn, and wheat based on the given resources and constraints for each scenario.

Step-by-step explanation:

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