A farming family has 100 hectares of arable land and $29,000 available for investment on the farm at the start of the current year.

Family members can produce a total of 3,500 hours of labor during the winter months (October to April) and 4,000 hours during the summer months. If labor hours are not needed during one season or another, family members can work as supernumeraries in a factory located a few kilometers away for $10/hour in winter or $12/hour in summer. . Annual income on the farm is generated by the cultivation of three cereals, namely wheat, corn and oats, and by the breeding of cattle and poultry.

In terms of investments, $300 per hectare will be required this year for wheat cultivation and $500 per hectare for corn cultivation regarding agricultural supplies. Furthermore, oats do not require any investment. As for animals, the purchase of a cattle is valued at $400 and each poultry costs $3 as an initial investment.

Each cattle requires 1.5 hectares of land and requires 100 hours of labor in winter and 50 hours in summer. Each cattle generates a net annual income of $3,700 (milk, meat). Poultry does not require significant land, requiring 0.6 hours of labor per unit in winter and 0.3 hours in summer for a net annual income of $30 (eggs, meat).

The henhouse can hold a maximum of 3000 poultry while the barn can accommodate a maximum of 32 cattle. Labor hours and net annual income per hectare cultivated for each cereal are as follows: The area dedicated to corn cultivation must represent at least 10% of the land allocated to all crops with the aim of primarily supplying the local market with corn cobs.

Finally, residual investments after allocation to cereals and farm animals can be invested in the financial markets at a forecast rate of 8%.

Questions :

1) The farming family must decide how many hectares will be allocated to each grain and how many of each type of farm animal will be purchased to maximize its net annual income. Help the family in its decision-making by formulating an appropriate linear model.

2) The family is offered the rental of part of its land (maximum 25 hectares) by a large farmer at the price of $10/hectare. Reformulate the model to take this proposition into account.

\begin{tabular}{|l|ccc|} \hline & Bl & Mas & Avoine \\ \hline Heures main d'uvre (hiver)/hectare & 20 & 35 & 10 \\ Heures main d'uvre (t)/hectare & 50 & 75 & 40 \\ Revenu annuel net (\$\$)/hectare & 2500 & 2300 & 800 \\ \hline \end{tabular}