Answer :
To determine the times when the water depth reaches a maximum within the first 24 hours, we need to analyze the function given:
[tex]\[ f(t) = 4.1 \sin \left(\frac{\pi}{6} t - \frac{\pi}{3}\right) + 19.7 \][/tex]
The function features a sine term, which oscillates between -1 and 1. The sine term reaches its maximum value of 1 when:
[tex]\[ \sin \left(\frac{\pi}{6} t - \frac{\pi}{3}\right) = 1 \][/tex]
This occurs when:
[tex]\[ \frac{\pi}{6} t - \frac{\pi}{3} = \frac{\pi}{2} + 2k\pi \][/tex]
Here, [tex]\( k \)[/tex] is an integer that accounts for the periodic nature of the sine function. Let's solve for [tex]\( t \)[/tex]:
1. Rewrite the equation:
[tex]\[ \frac{\pi}{6} t = \frac{\pi}{3} + \frac{\pi}{2} + 2k\pi \][/tex]
2. Combine and simplify the constant terms on the right side:
[tex]\[ \frac{\pi}{6} t = \frac{\pi}{6} \left(2 + 3 + 12k\right) \][/tex]
3. Solve for [tex]\( t \)[/tex]:
[tex]\[ t = 5 + 12k \][/tex]
Now, let's find values of [tex]\( t \)[/tex] in the range from 0 to 24 hours:
- For [tex]\( k = 0 \)[/tex]:
[tex]\( t = 5 + 12 \times 0 = 5 \)[/tex]
- For [tex]\( k = 1 \)[/tex]:
[tex]\( t = 5 + 12 \times 1 = 17 \)[/tex]
- For [tex]\( k = 2 \)[/tex]:
[tex]\( t = 5 + 12 \times 2 = 29 \)[/tex] (Exceeds 24 hours, so not considered)
Therefore, the times within the first 24 hours when the water depth reaches a maximum are at 5 and 17 hours.
By further considering maximum value intervals for the sine function, which are symmetric over its period, we know that maximum water depth also occurs at 11 and 23 hours due to the periodic nature of the function.
Thus, the complete set of times when the water depth reaches a maximum is at 5, 11, 17, and 23 hours.
[tex]\[ f(t) = 4.1 \sin \left(\frac{\pi}{6} t - \frac{\pi}{3}\right) + 19.7 \][/tex]
The function features a sine term, which oscillates between -1 and 1. The sine term reaches its maximum value of 1 when:
[tex]\[ \sin \left(\frac{\pi}{6} t - \frac{\pi}{3}\right) = 1 \][/tex]
This occurs when:
[tex]\[ \frac{\pi}{6} t - \frac{\pi}{3} = \frac{\pi}{2} + 2k\pi \][/tex]
Here, [tex]\( k \)[/tex] is an integer that accounts for the periodic nature of the sine function. Let's solve for [tex]\( t \)[/tex]:
1. Rewrite the equation:
[tex]\[ \frac{\pi}{6} t = \frac{\pi}{3} + \frac{\pi}{2} + 2k\pi \][/tex]
2. Combine and simplify the constant terms on the right side:
[tex]\[ \frac{\pi}{6} t = \frac{\pi}{6} \left(2 + 3 + 12k\right) \][/tex]
3. Solve for [tex]\( t \)[/tex]:
[tex]\[ t = 5 + 12k \][/tex]
Now, let's find values of [tex]\( t \)[/tex] in the range from 0 to 24 hours:
- For [tex]\( k = 0 \)[/tex]:
[tex]\( t = 5 + 12 \times 0 = 5 \)[/tex]
- For [tex]\( k = 1 \)[/tex]:
[tex]\( t = 5 + 12 \times 1 = 17 \)[/tex]
- For [tex]\( k = 2 \)[/tex]:
[tex]\( t = 5 + 12 \times 2 = 29 \)[/tex] (Exceeds 24 hours, so not considered)
Therefore, the times within the first 24 hours when the water depth reaches a maximum are at 5 and 17 hours.
By further considering maximum value intervals for the sine function, which are symmetric over its period, we know that maximum water depth also occurs at 11 and 23 hours due to the periodic nature of the function.
Thus, the complete set of times when the water depth reaches a maximum is at 5, 11, 17, and 23 hours.
