5 Findlay is trying to get into his local swimming team. The coach will watch him swim and will then make his decision. Findlay must swim at least two lengths using each stroke and must swim at least 8 lengths in total, taking at most 10 minutes.
Findlay needs to put together a routine that includes breaststroke, backstroke and butterfly. The table shows how Findlay expects to perform with each stroke.
| Stroke | Style marks | Time taken |
| Breaststroke | 2 marks per length | 2 minutes per length |
| Backstroke | 1 mark per length | 0.5 minutes per length |
| Butterfly | 5 marks per length | 1 minute per length |
Findlay needs to work out how many lengths to swim using each stroke to maximise his expected total number of style marks.
- Identify appropriate variables for Findlay's problem and write down the objective function, to be maximised, in terms of these variables.
- Formulate a constraint for the total number of lengths swum, a constraint for the time spent swimming and constraints on the number of lengths swum using each stroke.
Findlay decides that he will swim two lengths using butterfly. This reduces his problem to the following LP formulation:
$$\begin{array} { l c }
\text { maximise } & P = 2 x + y ,
\text { subject to } & x + y \geqslant 6 ,
& 4 x + y \leqslant 16 ,
& x \geqslant 2 , y \geqslant 2 ,
\end{array}$$
with \(x\) and \(y\) both integers. - Use a graphical method to identify the feasible region for this problem. Write down the coordinates of the vertices of the feasible region and hence find the integer values of \(x\) and \(y\) that maximise \(P\).
- Interpret your solution for Findlay.