| 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 \(p\).
Find the value of \(q\).
Use the model to find the depth of the water 10 hours after high tide.
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]
