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 is 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?
Ansewer
a) It is allowable for the farmer to grow some amount of each crop, as there are no restrictions mentioned in the problem scenario that limit the farmer to growing only one crop.
b) The farmer does not have to use all 1000 acres of land. The decision on how many acres of each crop to grow depends on factors such as the profitability of each crop and the available labor.
c) To determine the maximum number of acres of corn the farmer could grow, we need to consider the labor constraint. Each acre of corn requires 5 days of labor, and the farmer has 7500 days of labor available. Therefore, the maximum number of acres of corn the farmer could grow is 7500 divided by 5, which is 1500 acres. However, since the land available is only 1000 acres, the farmer can only grow a maximum of 1000 acres of corn. This will yield a profit of 1000 acres multiplied by $20, which is $20,000.
d) To find the maximum profit the farmer could make if he grew only wheat, we need to consider the labor constraint. Each acre of wheat requires 9 days of labor, and the farmer has 7500 days of labor available. Therefore, the maximum number of acres of wheat the farmer could grow is 7500 divided by 9, which is approximately 833. Since the farmer can only grow whole acres, we can round down to 833 acres. This will yield a profit of 833 acres multiplied by $40, which is $33,320.
e) If it were possible for the farmer to either obtain more land or cheaply hire more laborers, obtaining more land would be better. This is because the farmer already has enough available labor to fully utilize the existing land and maximize profits. Acquiring more land would allow for additional production and potentially higher profits, while hiring more laborers could lead to higher labor costs and reduce profitability.
a) It would be allowable for the farmer to grow some amount of each crop, as there are no constraints mentioned that would restrict the farmer to choose only one crop.
b) No, the farmer does not have to use all 1000 acres of his land. The farmer can choose to allocate any combination of corn, wheat, and soybeans based on profitability and available labor.
c) To determine the maximum number of acres of corn that the farmer could grow, we need to divide the total available days of labor (7500) by the number of days of labor required per acre of corn (5). Therefore, the maximum number of acres of corn that the farmer could grow is 7500/5 = 1500 acres.
The profit from growing corn would be the number of acres (1500) multiplied by the profit per acre ($20), which equals 1500 * 20 = $30,000.
d) To calculate the maximum profit if the farmer grew only wheat, we need to determine the maximum number of acres of wheat the farmer could grow. Using the same approach as above, the maximum number of acres of wheat would be 7500/9 = 833.33 acres. However, since the number of acres cannot be a fraction, the farmer can at most grow 833 acres of wheat.
The profit from growing wheat would be the number of acres (833) multiplied by the profit per acre ($40), which equals 833 * 40 = $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 farmer already has unused land available (1000 - 833 = 167 acres), so obtaining more land would not necessarily be beneficial.
By hiring more laborers, the farmer can increase the number of acres cultivated, leading to increased profitability. Additionally, hiring laborers may be more cost-effective compared to the cost of purchasing or leasing additional land.