Answer :
To find the times when the water depth in the harbor reaches its maximum during the first 24 hours, we need to analyze the provided function:
[tex]\[ f(t) = 4.1 \sin \left( \frac{\pi}{6} t - \frac{\pi}{3} \right) + 19.7 \][/tex]
Here's the detailed step-by-step solution:
1. Understanding the function:
- The given function is of the form [tex]\( f(t) = A \sin(B t - C) + D \)[/tex], where:
- [tex]\( A = 4.1 \)[/tex] is the amplitude.
- [tex]\( B = \frac{\pi}{6} \)[/tex] affects the period of the sine wave.
- [tex]\( C = \frac{\pi}{3} \)[/tex] is the phase shift.
- [tex]\( D = 19.7 \)[/tex] is the vertical shift.
2. Determining the period:
- The period of the sine function is given by [tex]\( \frac{2\pi}{B} \)[/tex].
- Here, [tex]\( B = \frac{\pi}{6} \)[/tex], so the period [tex]\( T = \frac{2\pi}{\frac{\pi}{6}} = 12 \)[/tex] hours.
3. Finding the maximum points:
- The sine function [tex]\( \sin(x) \)[/tex] reaches its maximum value at [tex]\( \sin(x) = 1 \)[/tex].
- Thus, we need:
[tex]\[ \frac{\pi}{6} t - \frac{\pi}{3} = \frac{\pi}{2} + 2n\pi \quad \text{for integer } n \][/tex]
- Simplifying this equation, we get:
[tex]\[ \frac{\pi}{6} t = \frac{\pi}{2} + \frac{\pi}{3} + 2n\pi \][/tex]
[tex]\[ \frac{\pi}{6} t = \frac{5\pi}{6} + 2n\pi \][/tex]
[tex]\[ t = 5 + 12n \][/tex]
- This means that the function reaches its maximum at times [tex]\( t = 5, 17, 29, \ldots \)[/tex]
4. Checking the times within the first 24 hours:
- From the expression [tex]\( t = 5 + 12n \)[/tex], the possible times during the first 24 hours (0 ≤ t ≤ 24) are:
- 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] (which is beyond the 24-hour period)
5. Conclusion:
- The water depth reaches its maximum at 5 hours and 17 hours during the first 24 hours.
Therefore, the times when the water depth reaches a maximum in the first 24 hours are:
- At 5 and 17 hours.
Thus, the correct answer is:
- At 5 and 17 hours.
[tex]\[ f(t) = 4.1 \sin \left( \frac{\pi}{6} t - \frac{\pi}{3} \right) + 19.7 \][/tex]
Here's the detailed step-by-step solution:
1. Understanding the function:
- The given function is of the form [tex]\( f(t) = A \sin(B t - C) + D \)[/tex], where:
- [tex]\( A = 4.1 \)[/tex] is the amplitude.
- [tex]\( B = \frac{\pi}{6} \)[/tex] affects the period of the sine wave.
- [tex]\( C = \frac{\pi}{3} \)[/tex] is the phase shift.
- [tex]\( D = 19.7 \)[/tex] is the vertical shift.
2. Determining the period:
- The period of the sine function is given by [tex]\( \frac{2\pi}{B} \)[/tex].
- Here, [tex]\( B = \frac{\pi}{6} \)[/tex], so the period [tex]\( T = \frac{2\pi}{\frac{\pi}{6}} = 12 \)[/tex] hours.
3. Finding the maximum points:
- The sine function [tex]\( \sin(x) \)[/tex] reaches its maximum value at [tex]\( \sin(x) = 1 \)[/tex].
- Thus, we need:
[tex]\[ \frac{\pi}{6} t - \frac{\pi}{3} = \frac{\pi}{2} + 2n\pi \quad \text{for integer } n \][/tex]
- Simplifying this equation, we get:
[tex]\[ \frac{\pi}{6} t = \frac{\pi}{2} + \frac{\pi}{3} + 2n\pi \][/tex]
[tex]\[ \frac{\pi}{6} t = \frac{5\pi}{6} + 2n\pi \][/tex]
[tex]\[ t = 5 + 12n \][/tex]
- This means that the function reaches its maximum at times [tex]\( t = 5, 17, 29, \ldots \)[/tex]
4. Checking the times within the first 24 hours:
- From the expression [tex]\( t = 5 + 12n \)[/tex], the possible times during the first 24 hours (0 ≤ t ≤ 24) are:
- 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] (which is beyond the 24-hour period)
5. Conclusion:
- The water depth reaches its maximum at 5 hours and 17 hours during the first 24 hours.
Therefore, the times when the water depth reaches a maximum in the first 24 hours are:
- At 5 and 17 hours.
Thus, the correct answer is:
- At 5 and 17 hours.
