6 Tamsin is planning how to spend a day off. She will divide her time between walking the coast path, visiting a bird sanctuary and visiting the garden centre.
Tamsin has given a value to each hour spent doing each activity. She wants to decide how much time to spend on each activity to maximise the total value of the activities.
| Activity | Walking coast path | Visiting bird sanctuary | Visiting garden centre |
| Value | 5 points per hour | 3 points per hour | 2 points per hour |
Tamsin's requirements are that she will spend:
- a total of exactly 6 hours on the three activities
- at most 3.5 hours walking the coast path
- at least as long at the bird sanctuary as at the garden centre
- at least 1 hour at the garden centre.
- If Tamsin spends \(x\) hours walking the coast path and \(y\) hours visiting the bird sanctuary, how many hours does she spend visiting the garden centre?
Tamsin models her problem as the linear programming formulation
Maximise \(P = 3 x + y\)
Subject to
$$\begin{aligned}
& x \leqslant 3.5
& x + 2 y \geqslant 6
& x + y \leqslant 5
& x \geqslant 0
\end{aligned}$$
and
- Explain why the maximum total value of the activities done is achieved when \(3 x + y\) is maximised.
- Show how the requirement that she spends at least as long at the bird sanctuary as at the garden centre leads to the constraint \(x + 2 y \geqslant 6\).
- Explain why there is no need to require that \(y \geqslant 0\).
Represent the constraints graphically and hence find a solution to Tamsin's problem.