The water depth in a harbor rises and falls over time. The function

[tex]f(t) = 4.1 \sin \left(\frac{\pi}{6} t - \frac{\pi}{3}\right) + 19.7[/tex]

models the water depth, in feet, after [tex]t[/tex] hours.

During the first 24 hours, at what times does the water depth reach a maximum?

A. at 5 and 17 hours
B. at 11 and 23 hours
C. at [tex]2, 8, 14[/tex], and 20 hours
D. at [tex]5, 11, 17[/tex], and 23 hours

To determine the times when the water depth in the harbor reaches a maximum, we need to analyze the function [tex]$ f(t) = 4.1 \sin \left(\frac{\pi}{6} t - \frac{\pi}{3}\right) + 19.7 $[/tex].

The expression [tex]$ \sin \left(\frac{\pi}{6} t - \frac{\pi}{3}\right) $[/tex] achieves its maximum value of 1. When the sine function reaches its maximum, the overall function [tex]$ f(t) $[/tex] will also be at a maximum, because we're adding a constant.

Let's solve for [tex]$ t $[/tex] when:

[tex]\[
\frac{\pi}{6} t - \frac{\pi}{3} = \frac{\pi}{2} + n\pi
\][/tex]

where [tex]$ n $[/tex] is an integer. This is because the sine function reaches its maximum at [tex]$ \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, $[/tex] etc.

Now, rearrange the equation to solve for [tex]$ t $[/tex]:

[tex]\[
\frac{\pi}{6} t = \frac{\pi}{2} + n\pi + \frac{\pi}{3}
\][/tex]

Combine the terms on the right:

[tex]\[
\frac{\pi}{6} t = \frac{3\pi}{6} + \frac{2\pi}{6} + n\pi = \frac{5\pi}{6} + n\pi
\][/tex]

Multiply both sides by 6 to solve for [tex]$ t $[/tex]:

[tex]\[
t = 5 + 6n
\][/tex]

Now, find the values of [tex]$ t $[/tex] for [tex]$ n $[/tex] that fall within the first 24 hours (where [tex]$ t \geq 0 $[/tex] and [tex]$ t < 24 $[/tex]):

1. For [tex]$ n = 0 $[/tex]: [tex]$ t = 5 $[/tex]
2. For [tex]$ n = 1 $[/tex]: [tex]$ t = 11 $[/tex]
3. For [tex]$ n = 2 $[/tex]: [tex]$ t = 17 $[/tex]
4. For [tex]$ n = 3 $[/tex]: [tex]$ t = 23 $[/tex]

These times are within the first 24 hours. Therefore, the water depth reaches a maximum at 5, 11, 17, and 23 hours.

So, the correct option is: at 5, 11, 17, and 23 hours.