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Date	May 2017	Marks available	2	Reference code	17M.2.sl.TZ2.4
Level	SL only	Paper	2	Time zone	TZ2
Command term	Find	Question number	4	Adapted from	N/A

Question

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]

Find the value of $q$ .

[2]

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

[2]

Markscheme

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]

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]

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]

Examiners report

[N/A]

Syllabus sections

Topic 3 - Circular functions and trigonometry » 3.4 » Composite functions of the form

$f(x) = a\sin (b(x + c)) + d$ .

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17M.2.sl.TZ2.4c: Use the model to find the depth of the water 10 hours after high tide.
17M.2.sl.TZ2.4a: Find the value of $p$ .

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