The depth of water in a port is modelled by the function d ( t ) = p cos ⁡ q t + 7.5, for 0 ⩽ t ⩽ 12, where t is the number of hours after high tide.At high tide, the depth is 9.7 metres.At low tide, which is 7 hours later, the depth is 5.3 metres.

Question

SLPaper 2

The depth of water in a port is modelled by the function $d(t) = p\cos qt + 7.5$ , for $0 \leqslant t \leqslant 12$ , where $t$ is the number of hours after high tide.

At high tide, the depth is 9.7 metres.

At low tide, which is 7 hours later, the depth is 5.3 metres.

Find the value of $p$ .

[2]

Verified

Solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

valid approach (M1)

eg $\,\,\,\,\,$ $\frac{{{\text{max}} - {\text{min}}}}{2}$ , sketch of graph, $9.7 = p\cos (0) + 7.5$

$p = 2.2$ A1 N2

[2 marks]

Find the value of $q$ .

[2]

Verified

Solution

valid approach (M1)

eg $\,\,\,\,\,$ $B = \frac{{2\pi }}{{{\text{period}}}}$ , period is $14,{\text{ }}\frac{{360}}{{14}},{\text{ }}5.3 = 2.2\cos 7q + 7.5$

0.448798

$q = \frac{{2\pi }}{{14}}{\text{ }}\left( {\frac{\pi }{7}} \right)$ , (do not accept degrees) A1 N2

[2 marks]

Use the model to find the depth of the water 10 hours after high tide.

[2]

Verified

Solution

valid approach (M1)

eg $\,\,\,\,\,$ $d(10),{\text{ }}2.2\cos \left( {\frac{{20\pi }}{{14}}} \right) + 7.5$

7.01045

7.01 (m) A1 N2

[2 marks]

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The depth of water in a port is modelled by the function d ( t ) = p cos ⁡ q t + 7.5, for 0 ⩽ t ⩽ 12, where t is the number of hours after high tide.At high tide, the depth is 9.7 metres.At low tide, which is 7 hours later, the depth is 5.3 metres.

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