Planning an excursion shuttle

An organiser must transport 86 pupils and 6 adults from a school to an activity centre. Everyone must be at the centre by 09:20. The group is ready at the school at 08:40. Each vehicle makes one outward trip; drivers are not included in passenger capacity. A mixed fleet from both companies is allowed. Vehicles may leave at different times; every passenger must meet the deadline.

The graph models one outward journey for every vehicle. Speed is in kilometres per hour and time is in minutes. Assume no extra stops or traffic delays for parts (a) and (b).

Any whole number of vehicles from either company is available at its listed time. Every selected vehicle is charged for one outward trip, even if not full. Boarding takes 8 minutes and starts only when both the group and vehicles are ready. Vehicles from each company can board simultaneously; boarding can also overlap between companies. Each vehicle may leave as soon as its own boarding is complete. Distance charges use the full outward distance from the graph. There are no return charges, taxes or other fees. Calculate each company’s total from unrounded values, then round the final total to the nearest cent.

1. Find the journey duration and distance travelled.[3]
2. Determine a minimum-cost transport plan that gets everyone to the centre by the deadline. Give the number of vehicles from each company, suitable departure and arrival times, and the total charge. Show that your choice meets the capacity and timing requirements and is cheaper than all other feasible fleet choices, or give a valid argument excluding more expensive choices.[10]
3. State one limitation of the journey model and explain its possible consequence for your plan.[1]