3 A farmer has 40 acres of land. Four crops, A, B, C and D are available.
Crop A will return a profit of \(\pounds 50\) per acre. Crop B will return a profit of \(\pounds 40\) per acre.
Crop C will return a profit of \(\pounds 40\) per acre. Crop D will return a profit of \(\pounds 30\) per acre.
The total number of acres used for crops A and B must not be greater than the total number used for crops C and D. The farmer formulates this problem as:
Maximise \(\quad 50 a + 40 b + 40 c + 30 d\),
subject to \(\quad a + b \leqslant 20\),
\(a + b + c + d \leqslant 40\).

1. Explain what the variables \(a , b , c\) and \(d\) represent. Explain how the first inequality was obtained.
   Explain why expressing the constraint on the total area of land as an inequality will lead to a solution in which all of the land is used.
2. Solve the problem using the simplex algorithm. Suppose now that the farmer had formulated the problem as:
   Maximise \(\quad 50 a + 40 b + 40 c + 30 d\),
   subject to \(\quad a + b \leqslant 20\),
   \(a + b + c + d = 40\).
3. Show how to adapt this problem for solution either by the two-stage simplex method or the big-M method. In either case you should show the initial tableau and describe what has to be done next. You should not attempt to solve the problem.