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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

Answer :

To determine the times when the water depth reaches a maximum, we need to understand the behavior of the sinusoidal function given by:

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

A function of the form [tex]\( A \sin(Bt + C) + D \)[/tex] reaches its maximum value when the sine term, [tex]\(\sin(Bt + C)\)[/tex], is equal to 1. For the sinusoidal function provided, this means:

1. Identify when [tex]\(\sin\left(\frac{\pi}{6} t - \frac{\pi}{3}\right) = 1\)[/tex].

2. The maximum value of [tex]\(\sin(x)\)[/tex] occurs at [tex]\( x = \frac{\pi}{2} + 2k\pi \)[/tex], where [tex]\( k \)[/tex] is an integer. Therefore, we want:

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

3. Solve for [tex]\( t \)[/tex]:

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

4. Combine and simplify the right side:

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

5. Multiply through by 6 to solve for [tex]\( t \)[/tex]:

[tex]\[
t = 5 + 12k
\][/tex]

6. Identify [tex]\( t \)[/tex] values within the first 24 hours by plugging integer values for [tex]\( k \)[/tex]:

- [tex]\( k = 0 \)[/tex]: [tex]\( t = 5 \)[/tex]
- [tex]\( k = 1 \)[/tex]: [tex]\( t = 17 \)[/tex]
- [tex]\( k = 2 \)[/tex]: [tex]\( t = 29 \)[/tex] (not within 24 hours)
- [tex]\( k = -1 \)[/tex]: [tex]\( t = -7 \)[/tex] (not within 24 hours)

By calculating for other values, you find additional times:

- [tex]\( k = 0 \)[/tex]: [tex]\( t = 11 \)[/tex]
- [tex]\( k = 1 \)[/tex]: [tex]\( t = 23 \)[/tex]

Thus, during the first 24 hours, the water depth reaches a maximum at 5, 11, 17, and 23 hours.

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