1. A farmer has 1000 acres of land on which corn, wheat, or soybeans can be grown. He also knows that each acre of corn requires 5 days of labor and yields a profit of $20. Each acre of wheat requires 9 days of labor and yields a profit of $40. Each acre of soybeans requires 10 days of labor and yields a profit of $50. The farmer has enough water for any of these crops and can count on his employees to supply as many as 7500 days of labor.
a) To demonstrate that you understand the situation, state whether you think it would be allowable for the farmer to grow some amount of each crop, or if the farmer must choose only one crop to grow.
b) Does the farmer have to use all 1000 acres of his land? Why or why not?
c) Based on the amount of labor that is available, what I s the maximum number of acres of corn that the farmer could grow? How much profit would that provide? [
d) What is the maximum profit the farmer could make if he grew only wheat?
e) If it were possible for the farmer to either obtain more land or to cheaply hire more laborers, which would be better? Why?
Extra Credit Suppose x represents the number of acres dedicated to corn, y represents the number of acres dedicated to wheat, and z represents the number of acres dedicated to soybeans. Give the profit function. Provide an inequality to represent that the farmer has no more that 1000 acres of land available. Give an inequality to represent that there is limited amount of available labor.
a) Based on the information provided, it is allowable for the farmer to grow some amount of each crop. There is no restriction mentioned that the farmer can only choose one crop to grow.
b) The farmer does not have to use all 1000 acres of his land. He can choose to use any combination of corn, wheat, and soybeans to make the best use of his resources.
c) To find the maximum number of acres of corn that the farmer could grow, we need to consider the labor available. Each acre of corn requires 5 days of labor, and the farmer has 7500 days of labor available. So, the maximum number of acres of corn can be calculated as 7500 divided by 5, which equals 1500 acres. However, since the farmer only has 1000 acres of land available, the maximum number of acres of corn the farmer could grow is 1000. The profit from growing corn can be calculated by multiplying the number of acres (1000) by the profit per acre ($20), which equals $20,000.
d) To find the maximum profit the farmer could make if he grew only wheat, we need to consider the labor available. Each acre of wheat requires 9 days of labor, and the farmer has 7500 days of labor available. So, the maximum number of acres of wheat can be calculated as 7500 divided by 9, which equals approximately 833. However, since the farmer only has 1000 acres of land available, the maximum number of acres of wheat the farmer could grow is 833. The profit from growing wheat can be calculated by multiplying the number of acres (833) by the profit per acre ($40), which equals $33,320.
e) If it were possible for the farmer to either obtain more land or to cheaply hire more laborers, it would be better to hire more laborers. This is because the profit per acre is higher for both wheat and soybeans compared to corn. By maximizing the labor input, the farmer can maximize the overall profit.
Extra Credit:
The profit function can be given as:
Profit = 20x + 40y + 50z
The inequality representing the availability of land can be written as:
x + y + z ≤ 1000
The inequality representing the availability of labor can be written as:
5x + 9y + 10z ≤ 7500