8. A company makes three sizes of lamps, small, medium and large. The company is trying to determine how many of each size to make in a day, in order to maximise its profit. As part of the process the lamps need to be sanded, painted, dried and polished. A single machine carries out these tasks and is available 24 hours per day. A small lamp requires one hour on this machine, a medium lamp 2 hours and a large lamp 4 hours. Let \(x =\) number of small lamps made per day, $$\begin{aligned} & y = \text { number of medium lamps made per day, }
& z = \text { number of large lamps made per day, } \end{aligned}$$ where \(x \geq 0 , y \geq 0\) and \(z \geq 0\).

1. Write the information about this machine as a constraint.
2. 1. Re-write your constraint from part (a) using a slack variable \(s\).
   2. Explain what \(s\) means in practical terms. Another constraint and the objective function give the following Simplex tableau. The profit \(P\) is stated in euros.

      Basic variable	\(x\)	\(y\)	\(z\)	\(r\)	\(s\)	Value
      \(r\)	3	5	6	1	0	50
      \(s\)	1	2	4	0	1	24
      \(P\)	- 1	- 3	- 4	0	0	0

3. Write down the profit on each small lamp.
4. Use the Simplex algorithm to solve this linear programming problem.
5. Explain why the solution to part (d) is not practical.
6. Find a practical solution which gives a profit of 30 euros. Verify that it is feasible.