The question asks for the time in hours until the next high tide. The depth of the water $d(t)$ in meters at a pier can be modeled by the equation:

$d(t) = 7 + 3 \cos\left(\frac{\pi}{6} t\right)$

where $t$ is the time in hours since the high tide. High tide occurs at 2:00 AM, one morning, and the next low tide occurs 6 hours later.

Steps to Solve:

1. Identify high tide conditions: High tide corresponds to the maximum depth of water. The maximum value of the cosine function is 1. Therefore, at high tide:

   $d(t) = 7 + 3 \times 1 = 10 \text{ meters}$

2. Find the next high tide: The equation for depth is periodic with period $T$. The cosine function completes one full cycle in $2\pi$ radians. For $\cos\left(\frac{\pi}{6} t\right)$, the period is:

   $\text{Period } T = \frac{2\pi}{\frac{\pi}{6}} = 12 \text{ hours}$

   Since the high tide occurs every 12 hours, if the first high tide is at 2:00 AM, the next one will occur 12 hours later at 2:00 PM.

Correct Answer:

Given the choices:

• A. 11:00 AM
• B. 12:00 PM
• C. 1:00 PM
• D. 2:00 PM

The correct answer is D. 2:00 PM.

Would you like more details or have any questions? Here are some related questions to explore further:

1. How would you determine the time of the next low tide?
2. What is the minimum depth of the water at the pier?
3. How would you calculate the time it takes for the depth to decrease from 10 meters to 7 meters?
4. How does the period of the function affect the timing of tides throughout the day?
5. What happens to the graph of the depth function if the amplitude of the cosine function is increased?

Tip: Understanding the properties of trigonometric functions, such as their periods and amplitudes, is crucial when dealing with problems involving periodic phenomena like tides.