Question 2 · Sinusoidal model: tides in a harbour
Parameters of $a\sin(b(t-c))+d$ from high and low tide; inequality with technology; limits of the model
Maximum mark · 18 marksThe depth of the water at a harbour entrance can be modelled by
where $h$ is the depth in metres and $t$ is the time in hours after midnight. On a particular day, high tide of $12.6$ m is recorded at 4:00 and the next low tide, of $5.4$ m, at 10:15.
- Find the values of $a$ and $d$, and interpret the meaning of each in the context of the problem. 4 p
- Find the period of the model and hence the values of $b$ and $c$. 5 p
- Find the depth of the water at 13:00. 3 p
- A ship can only enter the harbour if the depth of the water is at least $11$ m. Find the time interval, within the first $12$ hours of the day, during which the ship can enter. 4 p
- Explain one limitation of this model if it were used to predict the depth of the water a month from now. 2 p
Step-by-step solution
a) Amplitude and mean depth
$a$ is the amplitude: the greatest variation, in metres, from the mean depth. $d = 9$ m is the mean depth of the water.
Each value 1 mark, each interpretation 1 mark.
b) Period, $b$ and $c$
From high tide to low tide is half a period:
The maximum is at $t = 4$, so $b(4 - c) = \frac{\pi}{2}$ and therefore
Period 2 marks, $b$ 1 mark, $c$ 2 marks.
c) Depth at 13:00
Substitution 1 mark, calculator in radians 1 mark, answer 1 mark.
d) When the ship can enter
Using technology, $h(t) = 11$ at $t \approx 2.05$ and $t \approx 5.95$.
Setting up 2 marks, both solutions 1 mark, correct interval 1 mark.
e) One limitation
Any reasoned limitation. For example: the model assumes constant amplitude and period, whereas the real tidal range changes over the month with the phases of the moon (spring and neap tides); nor does it account for air pressure or wind.
Limitation 1 mark, justification in context 1 mark.