4 A farmer has 40 acres of land that can be used for growing wheat, potatoes and soya beans. The farmer can expect a profit of \(\pounds 80\) for each acre of wheat, \(\pounds 31\) for each acre of potatoes and \(\pounds 100\) for each acre of soya beans. Land that is left unplanted incurs no cost and generates no profit. The farmer wants to choose how much land to use for growing each crop to maximise the profit. It takes 4 hours to plant each acre of wheat, 2 hours to plant each acre of potatoes and 1 hour to plant each acre of soya beans. There are 60 hours available in total for planting. At most 25 acres can be used for wheat and at most 10 acres can be used for soya beans.
Let \(x\) denote the number of acres used for wheat, \(y\) denote the number of acres used for potatoes and \(z\) denote the number of acres used for soya beans.

1. Express the profit, \(\pounds P\), as a function of \(x , y\) and \(z\).
2. Explain why the constraint \(4 x + 2 y + z \leqslant 60\) is needed. Write down three more constraints on the values of \(x , y\) and \(z\), other than that they must be non-negative.
3. Set up an initial Simplex tableau to represent the farmer's problem. Perform one iteration of the Simplex algorithm, choosing a pivot from the column with the most negative value in the objective row. Show how each row that has changed was calculated. Julie uses the Simplex algorithm to solve the farmer's problem. Her final tableau is given below. The order of the rows and the use of the slack variables in Julie's tableau may be different from yours.

   P	\(x\)	\(y\)	\(z\)	\(s\)	\(t\)	\(u\)	\(v\)	RHS
   1	0	9	0	20	0	80	0	2000
   0	1	0.5	0	0.25	0	-0.25	0	12.5
   0	0	-0.5	0	-0.25	1	0.25	0	12.5
   0	0	0	1	0	0	1	0	10
   0	0	0.5	0	-0.25	0	-0.75	1	17.5

4. Write down the values of \(x , y\) and \(z\) from Julie's final tableau. Hence advise the farmer on how many acres to use for each crop and how much land should be left unplanted.