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 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 2, 8, 14, and 20 hours
D. at 5, 11, 17, and 23 hours

Answer :

To solve the problem of finding the times during the first 24 hours when the water depth reaches its maximum, we examine the function that models the water depth:

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

The water depth reaches its maximum when the sine function reaches its maximum value, which is 1. Therefore, we need to find when:

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

The sine function equals 1 at:

[tex]\[ \theta = \frac{\pi}{2} + 2k\pi \][/tex]

where [tex]\( k \)[/tex] is an integer representing the periodic nature of the sine function.

We set the argument of the sine function equal to these values to solve for [tex]\( t \)[/tex]:

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

Now, solve for [tex]\( t \)[/tex]:

1. Add [tex]\(\frac{\pi}{3}\)[/tex] to both sides:

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

2. Combine the fractions on the right side:

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

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

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

Therefore, the times [tex]\( t \)[/tex] within the first 24 hours are calculated by substituting different integer values for [tex]\( k \)[/tex]:

- For [tex]\( k = 0 \)[/tex]: [tex]\( t = 5 \)[/tex]
- For [tex]\( k = 1 \)[/tex]: [tex]\( t = 5 + 12 \times 1 = 17 \)[/tex]

Thus, the times when the water depth reaches its maximum during the first 24 hours are at 5 and 17 hours. None matches our expected times like 2, 8, 14, and 20 hours which is incorrect. The correct answer should be the result from the question choices.