The Bradley family owns 410 acres of farmland in North Carolina on which they gr

Question

The Bradley family owns 410 acres of farmland in North Carolina on which they grow corn and tobacco. Each acre of corn costs $105 to plant, cultivate, and harvest; each acre of tobacco costs $210. The Bradleys’ have a budget of $52,500 for next year. The government limits the number of acres of tobacco that can be planted to 100. The profit from each acre of corn is $300; the profit from each acre of tobacco is $520. The Bradleys’ want to know how many acres of each crop to plant in order to maximize their profit.

a. Formulate the linear programming model for the problem and solve.

b. How many acres of farmland will not be cultivated at the optimal solution? Do the Bradleys use the entire 100-acre tobacco allotment?

c. The Bradleys’ have an opportunity to lease some extra land from a neighbor. The neighbor is offering the land to them for $110 per acre. Should the Bradleys’ lease the land at that price? What is the maximum price the Bradleys’ should pay their neighbor for the land, and how much land should they lease at that price?

PLEASE USE EXCEL AND SHOW THE FORMULAS. ALSO WALK ME THROUGH HOW YOU GOT ALL THE ANSWERS

Bradley Family Let X1 = # of acres of corn and X2 = # of acres of tobacco Resources Corn Tobacco Available 300 1 105 0 520 Land Budget Govt. Restriction 410 52500 100 210 1 Government Restriction X1 X2 Decision Variables Objective function How many acres of farm land will not be cultivated at the optimal solution? Do the Bradleys use the entire 100 acre tobacco allotment? Should the Bradleys' lease extral land from their neighbor at $110 per acre? What is the maximum price the Bradleys' should pay their neighbor? And how much land shoud they lease? maximum price # of acres of land For each dollar they borrow, how much additional profit will they make? If they borrowed a additional $1000, would the number of acres of corn and tobacco they plant change?

Explanation / Answer

Let x and x are numbers of acres to grow corn/tobacco respectively, then the linear programming model is the following:
Maximize Profit = 300x + 520x,
subject to: x 0; 0 x 100 /Gov. limit/;
x + x 410 /Available farmland/;
105x + 210x 52500 /Budget restriction/

This can be solved either graphically or by simplex-method. The optimal solution is:
Maximum Profit = 300*320+500*90=$142 800; x = 320, x = 90.
Hence (320 + 90 = 410) there is no uncultivated land, 10 acres out of 100 for tobacco are unused.

Bradley Family (a) Let X1 = # of acres of corn and X2 = # of acres of tobacco Corn Tobacco 300 510 Land 1 1 320 <= 410 Budget 105 210 52500 <= 52500 Govt. Restriction 0 1 90 <= 100 Government Restriction X1 X2 Decision Variables 320 90 Objective Function 142800 (b) How many acres of farm land will not be cultivated at the optimal solution? 0 Do the Bradleys use the entire 100 acre tobacco allotment? no, only 90 (c) Should the Bradley's lease external land from their neighbor at $110 per acre? no What is the maximum price the Bradley's should pay their neighbor? And how much land should they lease? maximum price 80 # of acres of land 10 (d) For each dollar they borrow, how much additional profit will they make? 2.095 If they borrowed a additional $1000, would the number of the acres of corn and tobacco they plant change? yes

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