2024 ACS(BR) P1 Q12

A clock is set on the vertical wall on the roof of a building.

The distance from the centre of the clock, \(O\), to the tip, \(M\), of the minute hand is \(15\text{ m}\).

The height, \(h\text{ m}\), of \(M\) above the rooftop is given by \(h=a\cos kt+b\), where \(t\) is the time in minutes past the hour. \(O\) is \(20\text{ m}\) above the rooftop. The rooftop is taken to be horizontal.

1. Write down the value of \(a\) and explain why \(b=20\).[2]
2. Explain why the value of \(k\) is \(\dfrac\pi{30}\).[1]
3. Sketch the graph of \(h\) for \(0\le t\le45\).[3]
4. Find the two timings in the first hour for which the height of \(M\) is \(10\text{ m}\).[4]