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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 solve the problem of identifying when the water depth in the harbor reaches its maximum within the first 24 hours, we'll analyze the function:

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

### Step-by-Step Solution:

1. Understand the Sine Function:

The sine function, [tex]\(\sin(x)\)[/tex], reaches its maximum value of 1 at angles [tex]\(x = \frac{\pi}{2} + 2\pi n\)[/tex], where [tex]\(n\)[/tex] is an integer.

2. Set the Expression to Maximize Sine:

To find when the sine part of the water depth function is at its maximum, set:

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

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

Rearrange the equation and solve for [tex]\(t\)[/tex]:

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

Simplify the right-hand side:

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

Now substitute back:

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

Multiply both sides by [tex]\(\frac{6}{\pi}\)[/tex] to isolate [tex]\(t\)[/tex]:

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

This gives general times [tex]\(t = 5 + 12n\)[/tex], but we need to consider times within the first 24 hours.

4. Calculate Specific Times:

For [tex]\(n = 0\)[/tex], we get [tex]\(t = 5\)[/tex].

For [tex]\(n = 1\)[/tex], we get [tex]\(t = 17\)[/tex].

For [tex]\(n = 2\)[/tex], [tex]\(t = 29\)[/tex] which is outside the 24-hour range, so we don't include it.

5. Consider Cycle and Frequency:

The sine wave also has periodic maxima. With a period of [tex]\(\frac{12}{\pi} \times 2\pi = 12\)[/tex] hours, add 6 hours (half the period) to find intermediate maxima:

- [tex]\(5 + 6 = 11\)[/tex]
- [tex]\(17 + 6 = 23\)[/tex]

6. List All Maximum Times:

Compiling all valid times within the first 24 hours, we get: 5, 11, 17, and 23 hours.

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

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