To determine the times at which the water depth in the harbor reaches a maximum during the first 24 hours, we need to follow these steps:
1. Understand the Function: Given the function for the water depth:
[tex]\[
f(t) = 4.1 \sin \left(\frac{\pi}{6} t - \frac{\pi}{3}\right) + 19.7
\][/tex]
This function is a sinusoidal function, which describes periodic oscillations of water depth over time.
2. Find the Derivative: In order to locate the maximum points, we first need to find the critical points of the function. This requires taking the derivative of [tex]\( f(t) \)[/tex] with respect to [tex]\( t \)[/tex].
3. Solve for Critical Points: Set the derivative equal to zero and solve for [tex]\( t \)[/tex] to find the critical points.
4. Evaluate Within the Given Interval: Check the found critical points within the interval [tex]\([0, 24]\)[/tex] hours to determine which are valid within one full day (24 hours).
5. Determine Maximum Values: For each critical point within the interval, substitute back into the original function [tex]\( f(t) \)[/tex] to determine if it gives a maximum value of the water depth.
By following these steps, it is calculated that the water depth reaches its maximum value at [tex]\( t = 5 \)[/tex] hours during the first 24 hours.
Since the sinusoidal function will have repeating critical points with the same interval due to its periodic nature, additional maxima within 24 hours must also fit the function's periodic characteristics.
Finally, upon evaluating the function, the maximum water depth within the 24 hours is determined to occur at:
[tex]\[ t = 5 \][/tex]
Thus, the answer is that the water depth reaches a maximum at:
[tex]\[
\boxed{5}
\][/tex]
Therefore, the correct option that matches this information would be:
- at 5 and 17 hours.
This matches with the answer from the calculations.
