We are given the depth of water $d(t)$ in a harbor as a function of time $t$, modeled by:

$d(t) = 5 \cos \left( \frac{\pi}{6} t \right) + 7$

Where:

• $d(t)$ is the depth of water in meters.
• $t$ is the time in hours since midnight.

The problem asks when a boat that requires at least 6 meters of water to dock must leave the harbor and when it can return, assuming we need to find these times to the nearest minute.

Step 1: Set up the inequality for the boat to be able to dock

For the boat to dock, the depth $d(t)$ must be at least 6 meters. So we need to solve the inequality:

$5 \cos \left( \frac{\pi}{6} t \right) + 7 \geq 6$

Step 2: Simplify the inequality

First, subtract 7 from both sides:

$5 \cos \left( \frac{\pi}{6} t \right) \geq -1$

Now, divide by 5:

$\cos \left( \frac{\pi}{6} t \right) \geq -\frac{1}{5}$

Step 3: Solve for $t$

We now solve for $\frac{\pi}{6} t$. The cosine function achieves this value at specific angles. Using the inverse cosine:

$\frac{\pi}{6} t = \cos^{-1} \left( -\frac{1}{5} \right)$

This will give us the primary solution for $t$, but since cosine is periodic, we need to find all possible values of $t$ within a 24-hour period.

I will now calculate the specific time values numerically.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to guide you step-by-step through solving this manually or help with something else?

Here are five related questions that could help expand on this topic:

1. How can we determine the general solution of a trigonometric inequality?
2. How does the period of a cosine function affect the solution over a 24-hour period?
3. What are the steps for solving an equation involving the inverse cosine function?
4. How can we convert between radians and time when working with periodic functions?
5. How does the phase shift and amplitude affect the depth function in this problem?

Tip: Always remember to consider the periodic nature of trigonometric functions, especially when interpreting physical scenarios over time.