Answer :
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.
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.
