Trigonometry - Trigonometric Graphs and Parameters

The following diagram shows a waterwheel with a bucket. The wheel rotates at a constant rate in an anticlockwise (counterclockwise) direction.

The diameter of the wheel is \(8\) metres. The centre of the wheel, \(\mathrm{A}\), is \(2\) metres above the water level. After \(t\) seconds, the height of the bucket above the water level is given by \(h = a\sin{bt} +2\).

1. Show that \(a = 4\).[2]

The wheel turns at a rate of one rotation every \(30\) seconds.

2. Show that \(b = \frac{\pi}{15}\).[2]

In the first rotation, there are two values of \(t\) when the bucket is \(\textbf{descending}\) at a rate of \(0.5\) ms\(^{-1}\).

3. Find these values of \(t\).[6]
4. Determine whether the bucket is underwater at the second value of \(t\).[4]