<p>The following diagram shows a waterwheel with a bucket. The wheel rotates at a constant rate in an anticlockwise (counterclockwise) direction.</p>
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<p>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 \(h = a\sin{bt} +2\).</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that&nbsp;\(a = 4\).<span class="inline-marks">[2]</span></div></li></ol>
<p>The wheel turns at a rate of one rotation every \(30\) seconds.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Show that&nbsp;\(b = \frac{\pi}{15}\).<span class="inline-marks">[2]</span></div></li></ol>
<p>In the first rotation, there are two values of \(t\) when the bucket is \(\textbf{descending}\) at a rate of \(0.5\) ms\(^{-1}\).</p>
<ol class="abc-list" start="3"><li><div class="list-item-paragraph">Find these values of \(t\).<span class="inline-marks">[6]</span></div></li><li><div class="list-item-paragraph">Determine whether the bucket is underwater at the second value of \(t\).<span class="inline-marks">[4]</span></div></li></ol>

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 \(h = a\sin{bt} +2\).

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

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

Determine whether the bucket is underwater at the second value of \(t\).

Trigonometric Functions and Graphs

International Baccalaureate Standard Level Mathematics