7. A fitness centre runs introductory courses aimed at the following groups of customers: Pensioners, who will be charged \(\pounds 4\) for a 2 -hour session.
Other adults, who will be charged \(\pounds 10\) for a 4 -hour session.
Children, who will be charged \(\pounds 2\) for a 1 -hour session.
Let the number of pensioners, other adults, and children be \(x , y\) and \(z\) respectively.
Regulations state that the number of pensioners, \(x\), must be at most 5 more than the number of adults, \(y\). There must also be at least twice as many adults, \(y\), as there are children, \(z\). The centre is able to supervise up to 40 person-hours each day at the centre and wishes to maximise the revenue \(( \pounds R )\) that can be earned each day from these sessions. You may assume that the places on any courses that the centre runs will be filled.

1. Modelling this situation as a linear programming problem, write down the constraints and objective function in terms of \(x , y\) and \(z\). Using the Simplex algorithm, the following initial tableau is obtained.
2. \(\_\_\_\_\)