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 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.
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
