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.
[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.
