Question

Part A). 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 is $520. The Bradleys want to know how many acres of each crop to plant to maximize their profit.

Formulate a linear programming model for this problem.

Part B). Solve the linear programming model formulated in Part A for the Bradley family farm graphically.

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

b.) What would the profit for corn have to be for the Bradleys to plant only corn?

c.) If the Bradleys can obtain an additional 100 acres of land, will the number of acres of corn and tobacco they plan to grow change?

d.) If the Bradleys decide not to cultivate a 50-acre section as part of a crop recovery program, how will it affect their crop plans?

Part C). Solve the linear programming model formulated in Part A for the Bradley farm by using the computer.

A.) 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?

B.) The Bradleys are considering taking out a loan to increase their budget. For each dollar they borrow, how much additional profit would they make? If they borrowed an additional $1,000, would the number of acres of corn and tobacco they plant change?

Answer 1

Hi

So the answer of the following question is as follows :

Maximize Profit = 300x, + 520x2,

subject to: x; 2 0; 0S X S 100 /Gov. limit/;

X: + X; 3 410 /Available farmland/;

105x + 210x2 s 52500 /Budget restriction/

This can be solved either graphically or by simplex-method. The optimal solution is:

So the Maximum Profit = $142 800; X = 320, x 2 = 90. Hence (320 + 90 = 410) there is no uncultivated land, 10 acres out of 100 for tobacco are unused.

If only corn would be grown, add x2 = 0 to the above model, then optimal solution would be:

Maximum Profit = $123 000; x, = 410

=Next 100 extra acres, would require to change 410 to 510 in the above model, then optimal solution would be:

Maximum Profit = $150 000; x, = 500.

So the attached graph as follows :