=8-15 (Agricultural production planning problem) Margaret Black’s family owns five parcels of farmland broken into a southeast sector, north sector, northwest sector, west sector, and southwest sector. Margaret is involved primarily in growing wheat, alfalfa, and barley crops and is currently preparing her production plan for next year. The Pennsylvania Water Authority has just announced its yearly water allotment, with the Black farm receiving 7,400 acre-feet.

Each parcel can only tolerate a specified amount of irrigation per growing season, as specified in the following table:

Parcel Area (Acres)

Water Irrigation Limit

(Acre-Feet)

Southeast 2,000 3,200 North 2,300 3,400 Northwest 600 800 West 1,100 500 Southwest 500 600 Each of Margaret’s crops needs a minimum amount of water per acre, and there is a projected limit on sales of each crop. Crop data follow:

Crop Maximum Sales Water Needed Per Acre

(Acre-Feet)

Wheat 110,000 bushels 1.6 Alfalfa 1,800 tons 2.9 Barley 2,200 tons 3.5 Margaret’s best estimate is that she can sell wheat at a net profit of $2 per bushel, alfalfa at $40 per ton, and barley at $50 per ton. One acre of land yields an average of 1.5 tons of alfalfa and 2.2 tons of barley. The wheat yield is approximately 50 bushels per acre.

(a) Formulate Margaret’s production plan.

(b) What should the crop plan be, and what profit will it yield?

(c) The Water Authority informs Margaret that for a special fee of $6,000 this year, her farm will qualify for an additional allotment of 600 acrefeet of water. How should she respond?