We appreciate your visit to The water depth in a harbor rises and falls over time The function tex f t 4 1 sin left frac pi 6 t frac. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
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.
Thanks for taking the time to read The water depth in a harbor rises and falls over time The function tex f t 4 1 sin left frac pi 6 t frac. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
- Why do Businesses Exist Why does Starbucks Exist What Service does Starbucks Provide Really what is their product.
- The pattern of numbers below is an arithmetic sequence tex 14 24 34 44 54 ldots tex Which statement describes the recursive function used to..
- Morgan felt the need to streamline Edison Electric What changes did Morgan make.
Rewritten by : Barada
