Question

With a 102-acre farm on which Peaches and Apples are planted, you are asked to recommend the best allocation of each crop so that profits from these two crops are maximized. They want: - A maximum of 68 acres for Apples - At least 29 acres for Apples - At least 25 acres for Peaches >R represents the number of acres allocated for peaches and, S represents the number of acres allocated for apples. Now: Management is considering leasing additional 23 acres of land so that the two crops can be planted on 125 acres, instead of 102 acres. Assume that the profits per acre of Peaches and Apples are equal, say, P, where P>0.

(1) How does this change affect the formulation of the problem? does the objective function change? the feasible region?
(2) What would be the effect on the optimal allocation of acres for each crop, if there is any effect? Specify the optimal allocation of acres for Peaches and Apples if the farm has 125 acres
(3) What would be the effect on the optimal profit? Express the answer as a function of P.
(4)|What is the maximum they should be willing to pay for each additional acre? Express the answer as a function of P

Answer 1

The change in land size affects the feasible region and not the objective function. The optimal allocation and profit can be determined using linear programming with the updated constraints. The maximum willingness to pay for each additional acre depends on the profit per acre and the optimal allocation.

Step-by-step explanation:

The change of leasing an additional 23 acres of land from the 102-acre farm to make it 125 acres affects the formulation of the problem by expanding the feasible region. The objective function remains the same as it is still aimed at maximizing profits. The best allocation of acres for peaches and apples on the 125-acre farm can be determined by solving the linear programming problem using the updated constraints.

The optimal allocation of acres for peaches and apples on the 125-acre farm will depend on the specific values assigned to each crop in terms of maximizing profits. To determine this optimal allocation, the linear programming problem needs to be solved using the updated constraints.

The effect on the optimal profit can be expressed as a function of P, where P represents the profit per acre of peaches and apples. The specific value of this function will depend on the values assigned to P and the optimal allocation of acres for each crop.

The maximum the farm should be willing to pay for each additional acre can be expressed as a function of P. The specific value of this function will depend on the values assigned to P and the optimal allocation of acres for each crop.