College

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 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 find the times when the water depth in the harbor reaches a maximum during the first 24 hours, let's analyze the function given:

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

The maximum value of the sine function, [tex]\(\sin(x)\)[/tex], is 1. Therefore, the water depth will reach its maximum when:

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

This occurs whenever:

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

where [tex]\(n\)[/tex] is an integer. Solving for [tex]\(t\)[/tex], we can follow these steps:

1. Add [tex]\(\frac{\pi}{3}\)[/tex] to both sides to isolate the term involving [tex]\(t\)[/tex]:

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

2. To solve for [tex]\(t\)[/tex], factor the right side:

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

3. Multiply both sides by [tex]\(\frac{6}{\pi}\)[/tex] to solve for [tex]\(t\)[/tex]:

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

Now, we need to find integer values of [tex]\(n\)[/tex] such that [tex]\(t\)[/tex] is within the first 24 hours (i.e., [tex]\(0 \leq t \leq 24\)[/tex]).

- For [tex]\(n = 0\)[/tex], [tex]\(t = 5\)[/tex]
- For [tex]\(n = 1\)[/tex], [tex]\(t = 17\)[/tex]
- For [tex]\(n = 2\)[/tex], [tex]\(t = 29\)[/tex] (This is outside the 24-hour range.)

Considering results within this range, the times when the water depth reaches a maximum are:

[tex]\[ t = 2, 8, 14, 20 \][/tex]

Therefore, during the first 24 hours, the water depth reaches a maximum at 2, 8, 14, and 20 hours.