https://file.aiquickdraw.com/mathbot/file/1726192632828zoq3ecns.jpg

Let's analyze the problem provided in the image.

We are given a sinusoidal model of the height of sea water as:

\[
h(t) = p \sin(qt) + r
\]
where:
- \( t \) is the time in hours after 23:00 on 7 April 2018,
- \( p \), \( q \), and \( r \) are constants.

Points \( A(8.25, 0.4) \) and \( B(13.75, 1.8) \) represent the first low tide and the next high tide, respectively.

Now, let's solve part (a):

### (a) (i) How much time is there between the first low tide and the next high tide?

The time for the first low tide is 8.25 hours and the time for the next high tide is 13.75 hours. The difference in time between these two events is:

\[
13.75 - 8.25 = 5.5 \text{ hours}
\]

Thus, the time difference between the first low tide and the next high tide is **5.5 hours**.

### (a) (ii) Find the difference in height between the low tide and high tide.

The height at the first low tide is 0.4 meters, and the height at the next high tide is 1.8 meters. The difference in height is:

\[
1.8 - 0.4 = 1.4 \text{ meters}
\]

So, the difference in height between the low tide and the high tide is **1.4 meters**.

---

Would you like a deeper explanation or have any further questions? Here are some related questions for further exploration:

1. How can we determine the constants \( p \), \( q \), and \( r \) in the function \( h(t) \)?
2. What is the period of the sinusoidal function \( h(t) \)?
3. How could we model the time for the next low tide using this equation?
4. What would be the height of the sea water 24 hours after the first low tide?
5. How can we calculate the maximum and minimum values of the function \( h(t) \)?

**Tip:** To solve sinusoidal equations like this, it's useful to remember that \( \sin(t) \) has a period of \( 2\pi \), which helps in determining the cycle of tides.

In a coast, the height of the sea water in metres is modeled by the function h(t) = p sin(qt) + r, where t is the number of hours after 23:00 hours on 7 April 2018, for 0 ≤ t ≤ 72. The point A(8.25, 0.4) represents the first low tide and B(13.75, 1.8) represents the next high tide.
(a) (i) How much time is there between the first low tide and the next high tide?
(ii) Find the difference in height between the low tide and high tide.

Learn how to solve problems involving sinusoidal models with a focus on calculating the time difference between low and high tides and finding the height difference using trigonometric functions. Suitable for students in Grades 10-12.

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