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    <name>
      <text>5.3.2.7 Example</text>
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      <text><![CDATA[<hr>
<h4 class="HELM_example">Example</h4>
<p>The diagram shows a typical variation of the depth (\(d\) metres) of water in a particular harbour with time (\(t\) hours) as the depth changes with the tide.</p>
<p><img src="@@PLUGINFILE@@/5-3-tide_depth_time.svg" alt="" width="335" height="245" role="presentation" class="img-responsive atto_image_button_text-bottom"><br></p>

<ol class="HELM_parts">
    <li>Find a suitable equation for the curve in the diagram</li>
    <li>A boat enters the harbour in late morning on a day when the high tide is at 2 p.m. The boat needs a water depth of \(4\,\textrm{m}\) to sail safely. What advice would you give to its pilot about when to leave the harbour if the boat is not to be forced
        to wait in the harbour through the evening low tide?
    </li>
    <li>State two modelling assumptions you have made</li>
</ol>

<h4 class="HELM_solution">Solution</h4>
<ol class="HELM_parts">
    <li>
        <p>The equation is of the form</p>
        <p>\( \quad h = a+b\cos(\omega t) \quad \) (or \(h= a+b\sin \left(\omega t + \frac{\pi}{2}\right)\) )</p>
        <p>By inspection, \( a= 5 \) and \(b=3.\)</p>
        <p>The period \( T = 12.5 = \dfrac{2 \pi}{\omega}\) so \(\omega = \dfrac{4 \pi}{25} = 0.502655\)</p>
        <p>so the equation of the curve is \(h = 5+3 \cos \left(\dfrac{4 \pi}{25}t \right).\)</p>
    </li>

    <li>
        <p>Put \( h= 4 \) into the equation:</p>
        <p>\( \quad 4 = 5+3\cos \left(\dfrac{4 \pi}{25}t \right) \) implying \( -\dfrac{1}{3}= \cos\left(\dfrac{4 \pi}{25}t \right) \)</p>
        <p>Now, inverting the cosine:</p>
        <p>\( \dfrac{4 \pi}{25}t = \cos^{-1} \left(-\dfrac{1}{3}\right) = 1.91063\) giving \(t = 3.80108\) hours.</p>
        <p>So the advice to the pilot should be that he needs to be clear of the harbour by 5:45 pm at the very latest - and that he should allow a safety margin.</p>
    </li>
  
  <li><p>Assumptions likely are:</p>
    <p>The tide on the day in question is typical.</p>
    <p>No waves.</p><p>
    </p><p>A sinusoidal function accurately models the effect of the tide on sea level.</p>
  </li>
</ol>]]></text>
<file name="5-3-tide_depth_time.svg" path="/" encoding="base64">[encoded data, 11984 characters]</file>
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