2025 AHS PRELIMS P2 Q7

A waterwheel rotates at 1 revolution per 12 seconds. Initially, Point \(A\), marked on the highest point of the rim of the wheel is 6 m above the water surface. It reaches 2 m below the water surface 6 seconds later.

The motion of point \(A\) can be modelled using the trigonometric equation \(h=b\cos(kt)+2\), where the height of point \(A\) above the water surface, \(h\), is a function of time \(t\), in seconds.

1. Show that \(b=4\) and \(k=\frac{\pi}{6}\).[2]
2. On the axes below, sketch the graph of \(h\), for \(0\leq t\leq18\).[3]

A similar waterwheel at a different location has a point, \(B\), marked on its rim. The motion of point \(B\) is modelled by the equation \(h_B=4\cos\left(\frac{\pi}{6}t\right)+4\).

3. By drawing a suitable straight line on the same axes above, determine the number of time(s) when point \(B\) touches the water surface from \(0\leq t\leq18\).[2]