2025 HCI P1 Q5

The tourism board plans to construct a tram track around a tourist attraction. The diagram above shows part of the blueprint of the track, where the tram will run along a circular path with centre \(O\) and radius \(450\) m, from a fixed point \(A\) to a variable point \(B\), and then straight across to a fixed point \(C\). The angle \(AOB\) is denoted as \(\theta\), where \(0 \le \theta \le \pi\). The speed of the tram along the circular path will be maintained at \(8\) m/s and the speed along the straight path \(BC\) will be maintained at \(\sqrt{6}\) m/s.

1. Show that the time taken for the tram to travel from point \(A\) to \(C\) is \[\frac{225}{4} \theta + 150\sqrt{6} \cos \frac{\theta}{2}\][3]
2. Use calculus to find the maximum time taken for the tram to travel from point \(A\) to \(C\), giving your answer correct to the nearest minute. (You need not show that your answer gives a maximum.)[3]