Harbour depth and the window for a ship
Harbour depth and the window for a ship
IP 4
10 marks
The depth of water at a harbour entrance, \(d\) metres, \(t\) hours after midnight, is modelled by \(d=5+2\sin\left(\dfrac{\pi t}{6}\right)\) for \(0\le t\le12\).
- Find the rate at which the depth is changing at 2 a.m.[2]
- Find the times in \(0\le t\le12\) at which the depth is changing fastest, stating whether the water is rising or falling at each time.[3]
- A ship needs a depth of at least \(6\) m. Find the period during which it can use the entrance.[3]
- Real tides repeat about every \(12.4\) hours. Explain how this affects the model's prediction for the next day if the same formula is used for \(12\le t\le24\).[2]
Solution:
Solution locked
680+ questions here come with free worked solutions. A free account saves your questions and progress; this one needs tuition enrolment.
Create a free account to practise similar questions.
Answer:(a) \(\dfrac{\pi}{6}\approx0.524\) m per hour (b) \(t=0\) and \(t=12\) (rising), \(t=6\) (falling) (c) \(1\le t\le5\), 1 a.m. to 5 a.m. (d) the real tide drifts about \(24\) minutes later each cycle, so the prediction becomes increasingly early
Need help? Join our IP Math tuition programme.
Learn more