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Answer :
To determine when the water depth reaches a maximum during the first 24 hours, we look at the function [tex]\( f(t) = 4.1 \sin \left(\frac{\pi}{6} t - \frac{\pi}{3}\right) + 19.7 \)[/tex].
Here's a step-by-step explanation:
1. Understanding the Function's Behavior:
- The term [tex]\( \sin \left(\frac{\pi}{6} t - \frac{\pi}{3}\right) \)[/tex] represents a sinusoidal function that oscillates between -1 and 1.
- The coefficient 4.1 scales this sine wave, and adding 19.7 shifts it upwards, setting the water depth baseline.
2. Finding the Maximum Points:
- The maximum value of the sine function, [tex]\(\sin(\text{anything})\)[/tex], is 1.
- Therefore, [tex]\( f(t) \)[/tex] achieves its maximum when [tex]\( \sin \left(\frac{\pi}{6} t - \frac{\pi}{3}\right) = 1 \)[/tex].
3. Setting Up the Equation:
- The general solution to [tex]\(\sin(\theta) = 1\)[/tex] occurs at [tex]\(\theta = \frac{\pi}{2} + 2n\pi\)[/tex], where [tex]\( n \)[/tex] is an integer.
- Substituting in our problem: [tex]\(\frac{\pi}{6} t - \frac{\pi}{3} = \frac{\pi}{2} + 2n\pi\)[/tex].
4. Solving for [tex]\( t \)[/tex]:
- Rearrange the equation: [tex]\(\frac{\pi}{6} t = \frac{\pi}{2} + \frac{\pi}{3} + 2n\pi\)[/tex].
- Combine like terms: [tex]\(\frac{\pi}{6} t = \frac{5\pi}{6} + 2n\pi\)[/tex].
- Solve for [tex]\( t \)[/tex]: [tex]\( t = 5 + 12n \)[/tex].
5. Finding Specific Times Within 24 Hours:
- Substitute different integer values for [tex]\( n \)[/tex] to get:
- For [tex]\( n = 0 \)[/tex], [tex]\( t = 5 + 12 \times 0 = 5 \)[/tex].
- For [tex]\( n = 1 \)[/tex], [tex]\( t = 5 + 12 \times 1 = 17 \)[/tex].
Thus, during the first 24 hours, the water depth reaches a maximum at 5 and 17 hours. There might be a misunderstanding in showing the output from previous analysis, but based on the correct understanding of the sinusoidal function, these are the times the water depth reaches its maximum. Therefore, the correct option is "at 5 and 17 hours."
Here's a step-by-step explanation:
1. Understanding the Function's Behavior:
- The term [tex]\( \sin \left(\frac{\pi}{6} t - \frac{\pi}{3}\right) \)[/tex] represents a sinusoidal function that oscillates between -1 and 1.
- The coefficient 4.1 scales this sine wave, and adding 19.7 shifts it upwards, setting the water depth baseline.
2. Finding the Maximum Points:
- The maximum value of the sine function, [tex]\(\sin(\text{anything})\)[/tex], is 1.
- Therefore, [tex]\( f(t) \)[/tex] achieves its maximum when [tex]\( \sin \left(\frac{\pi}{6} t - \frac{\pi}{3}\right) = 1 \)[/tex].
3. Setting Up the Equation:
- The general solution to [tex]\(\sin(\theta) = 1\)[/tex] occurs at [tex]\(\theta = \frac{\pi}{2} + 2n\pi\)[/tex], where [tex]\( n \)[/tex] is an integer.
- Substituting in our problem: [tex]\(\frac{\pi}{6} t - \frac{\pi}{3} = \frac{\pi}{2} + 2n\pi\)[/tex].
4. Solving for [tex]\( t \)[/tex]:
- Rearrange the equation: [tex]\(\frac{\pi}{6} t = \frac{\pi}{2} + \frac{\pi}{3} + 2n\pi\)[/tex].
- Combine like terms: [tex]\(\frac{\pi}{6} t = \frac{5\pi}{6} + 2n\pi\)[/tex].
- Solve for [tex]\( t \)[/tex]: [tex]\( t = 5 + 12n \)[/tex].
5. Finding Specific Times Within 24 Hours:
- Substitute different integer values for [tex]\( n \)[/tex] to get:
- For [tex]\( n = 0 \)[/tex], [tex]\( t = 5 + 12 \times 0 = 5 \)[/tex].
- For [tex]\( n = 1 \)[/tex], [tex]\( t = 5 + 12 \times 1 = 17 \)[/tex].
Thus, during the first 24 hours, the water depth reaches a maximum at 5 and 17 hours. There might be a misunderstanding in showing the output from previous analysis, but based on the correct understanding of the sinusoidal function, these are the times the water depth reaches its maximum. Therefore, the correct option is "at 5 and 17 hours."
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