This problem involves the equation for the depth of water at the entrance of a harbor, given by:

$h(t) = 8 + 4 \sin\left(\frac{\pi t}{6}\right)$

where $h(t)$ is the depth of the water in meters, and $t$ is the time in hours after midnight (ranging from 0 to 24 hours).

Let's now proceed to the given parts of the problem.

(a) Find the maximum depth and the minimum depth of the water.

• The function $h(t) = 8 + 4 \sin\left(\frac{\pi t}{6}\right)$ involves a sine function.
• The sine function oscillates between -1 and 1. Hence, $\sin\left(\frac{\pi t}{6}\right)$ will also oscillate between -1 and 1.

Therefore:

• The maximum value of $\sin\left(\frac{\pi t}{6}\right)$ is 1, leading to: $h_{\text{max}} = 8 + 4(1) = 12 \text{ meters}.$

• The minimum value of $\sin\left(\frac{\pi t}{6}\right)$ is -1, leading to: $h_{\text{min}} = 8 + 4(-1) = 4 \text{ meters}.$

Thus, the maximum depth of water is 12 meters, and the minimum depth of water is 4 meters.

(b) Find the values of $t$ for which $h(t) \geq 8$.

We are asked to find the times when the depth of water is greater than or equal to 8 meters. This implies:

$h(t) \geq 8$

Substitute the expression for $h(t)$:

$8 + 4 \sin\left(\frac{\pi t}{6}\right) \geq 8$

Subtract 8 from both sides:

$4 \sin\left(\frac{\pi t}{6}\right) \geq 0$

Now, divide by 4:

$\sin\left(\frac{\pi t}{6}\right) \geq 0$

The sine function is non-negative when its argument is between 0 and $\pi$, which happens for values of $t$ where:

$0 \leq \frac{\pi t}{6} \leq \pi$

Multiply both sides by 6:

$0 \leq t \leq 6$

Hence, $h(t) \geq 8$ for $t$ in the range $[0, 6]$ hours. The same condition holds for $t$ in the range $[12, 18]$ hours due to the periodicity of the sine function.

Thus, the values of $t$ for which $h(t) \geq 8$ are $0 \leq t \leq 6$ and $12 \leq t \leq 18$.

Would you like further details on the steps, or do you have any questions about the solution? Here are some related questions for deeper understanding:

1. How does the sine function's periodicity affect the depth of water over time?
2. Can you calculate the value of $h(t)$ at any specific time, such as $t = 3$ hours?
3. What would happen to the depth if the coefficient of the sine function were increased or decreased?
4. How could this function be applied to model tides or ocean waves?
5. Can you explore the behavior of the depth outside the time interval [0, 24] hours?

Tip: The sine function's period is $2\pi$, which means it repeats its values every 2π units. Adjusting coefficients inside the sine function alters its frequency and amplitude.