Let p_1,p_2 "and" p_3 be the acres of land that will be planted for soybean, corn and wheat, respectively. Let q_1 "and" q_2 be the number of additional cows and hens, respectively, that the farm should buy. Note that the q_i's must be both integers while the p_i's need not be integers.

At the end the of the year, the farm has 30+q_1 cows and 2000+q_2 hens which generated a net cash income of

850(30+q_1 )+4.25(2000+q_2 )=34000+850q_1+4.25q_2

The crops, on the other hand, the crop’s net value (either in net income or savings in purchasing feed for the livestock) will be

70p_1+60p_2+40p_3.

At the end of the year, after incorporating the aging of the livestock, the value of the livestock will be

0.9(35000+1500q_1 )+0.75(5000+3q_2 )=35250+1350q_1+2.25q_2

Next, we calculate the income of the entire family from working outside their farm if they will have unused person-hours. During the winter and spring time, the total hours that they will work in the their farm will be

p_1+0.9p_2+0.6p_3+10(6)(30+q_1 )+0.05(6)(2000+q_2 )=p_1+0.9p_2+0.6p_3+60q_1+0.3q_2+2400

While during summer and fall, the total hours that they will work in their farm will be

1.4p_1+1.2p_2+0.7p_3+10(6)(30+q_1 )+0.05(6)(2000+q_2 )=1.4p_1+1.2p_2+0.7p_3+60q_1+0.3q_2+2400

Thus provided that these total hours do not exceed the available total hours for the season, then their total income from working from outside their farm will be

5(4000-p_1-0.9p_2-0.6p_3-60q_1-0.3q_2-2400)+5.5(4500-1.4p_1-1.2p_2-0.7p_3-60q_1-0.3q_2-2400)=19550-12.7p_1-11.1p_2-6.85p_3-630q_1-3.15q_2

Incorporating now the cost of the new livestock and the living expenses per year, then the net worth P of the family at the end of the year will be

P=20000-1500q_1-3q_2-40000+34000+850q_1+4.25q_2+70p_1+60p_2+40p_3+35250+1350q_1+2.25q_2+19550-12.7p_1-11.1p_2-6.85p_3-630q_1-3.15q_2
=68800+70q_1+0.35q_2+57.3p_1+48.9p_2+33.15p_3

Problem 1.3: The following transportation problem is given: Plant...