14 Douglas wants to construct a model for the height of the tide in Liverpool during the day, using a cosine graph to represent the way the height changes.
He knows that the first high tide of the day measures 8.55 m and the first low tide of the day measures 1.75 m .
Douglas uses \(t\) for time and \(h\) for the height of the tide in metres. With his graph-drawing software set to degrees, he begins by drawing the graph of \(\mathrm { h } = 5.15 + 3.4\) cost.
- Verify that this equation gives the correct values of \(h\) for the high and low tide.
Douglas also knows that the first high tide of the day occurs at 1 am and the first low tide occurs at 7.20 am. He wants \(t\) to represent the time in hours after midnight, so he modifies his equation to \(h = 5.15 + 3.4 \cos ( a t + b )\).
- Show that Douglas's modified equation gives the first high tide of the day occurring at the correct time if \(\mathrm { a } + \mathrm { b } = 0\).
- Use the time of the first low tide of the day to form a second equation relating \(a\) and \(b\).
- Hence show that \(a = 28.42\) correct to 2 decimal places.
- Douglas can only sail his boat when the height of the tide is at least 3 m .
Use the model to predict the range of times that morning when he cannot sail.
- The next high tide occurs at 12.59 pm when the height of the tide is 8.91 m .
Comment on the suitability of Douglas's model.