Week 4: Trig Modelling Master Class

The Context:

A large cargo ship, the Maersk Nelson, needs to enter the harbour. The depth of the water in the channel changes with the tide.

The Data:

Low Tide: The water is 1.8 meters deep at 9:45 AM.
High Tide: The water is 4.2 meters deep at 3:30 AM (and again at 4:00 PM).
The tidal cycle (Period) is approximately 12.5 hours.

The Task:

Before modelling, we must define the variables. This changes the value of $B$!

1. Let $y$ represent:

2. Let $x$ (or $t$) represent:

We use the form: $y = A \cos(B(x-C)) + D$

Amplitude (A): Use $A = \frac{Max - Min}{2}$

Vertical Shift (D): Use $D = \frac{Max + Min}{2}$ (Average depth)

Frequency (B): Period is 12.5 hours. Use $B = \frac{2\pi}{Period}$

Horizontal Shift (C): At what time ($x$) is the first High Tide (Peak)?

(High Tide is at 3:30 AM. Convert 3:30 into decimal hours)

Does your model match the data points?

The ship needs 2.5m depth.

1. In Desmos above, add the line $y=2.5$.

2. Find the first time the water drops below 2.5m (Intersection 1).

First unsafe time (hours):

The harbour master wants to know the total duration (in hours) during the daylight hours (6:00 AM to 6:00 PM) that the ship can safely enter.

1. Graph the function $y = 1.2\cos(0.503(x-3.5)) + 3$.

2. Graph the constraint $y > 2.5$.

3. Restrict domain: $\{6 < x < 18\}$.

Question: Calculate the total hours available between 0600 and 1800.