5 John is reviewing his lifestyle, and in particular his leisure commitments. He enjoys badminton and squash, but is not sure whether he should persist with one or both. Both cost money and both take time.

Playing badminton costs $\pounds 3$ per hour and playing squash costs $\pounds 4$ per hour. John has $\pounds 11$ per week to spend on these activities.

John takes 0.5 hours to recover from every hour of badminton and 0.75 hours to recover from every hour of squash. He has 5 hours in total available per week to play and recover.\\
(i) Define appropriate variables and formulate two inequalities to model John's constraints.\\
(ii) Draw a graph to represent your inequalities.

Give the coordinates of the vertices of your feasible region.\\
(iii) John is not sure how to define an objective function for his problem, but he says that he likes squash "twice as much" as badminton. Letting every hour of badminton be worth one "satisfaction point" define an objective function for John's problem, taking into account his "twice as much" statement.\\
(iv) Solve the resulting LP problem.\\
(v) Given that badminton and squash courts are charged by the hour, explain why the solution to the LP is not a feasible solution to John's practical problem. Give the best feasible solution.\\
(vi) If instead John had said that he liked badminton more than squash, what would have been his best feasible solution?

\hfill \mbox{\textit{OCR MEI D1 2006 Q5 [16]}}