Answer :
To determine the times when the water depth reaches a maximum, we need to understand the behavior of the sinusoidal function given by:
[tex]\[ f(t) = 4.1 \sin\left(\frac{\pi}{6} t - \frac{\pi}{3}\right) + 19.7 \][/tex]
A function of the form [tex]\( A \sin(Bt + C) + D \)[/tex] reaches its maximum value when the sine term, [tex]\(\sin(Bt + C)\)[/tex], is equal to 1. For the sinusoidal function provided, this means:
1. Identify when [tex]\(\sin\left(\frac{\pi}{6} t - \frac{\pi}{3}\right) = 1\)[/tex].
2. The maximum value of [tex]\(\sin(x)\)[/tex] occurs at [tex]\( x = \frac{\pi}{2} + 2k\pi \)[/tex], where [tex]\( k \)[/tex] is an integer. Therefore, we want:
[tex]\[
\frac{\pi}{6} t - \frac{\pi}{3} = \frac{\pi}{2} + 2k\pi
\][/tex]
3. Solve for [tex]\( t \)[/tex]:
[tex]\[
\frac{\pi}{6} t = \frac{\pi}{2} + 2k\pi + \frac{\pi}{3}
\][/tex]
4. Combine and simplify the right side:
[tex]\[
\frac{\pi}{6} t = \frac{\pi}{2} + \frac{\pi}{3} + 2k\pi = \frac{3\pi}{6} + \frac{2\pi}{6} + 2k\pi = \frac{5\pi}{6} + 2k\pi
\][/tex]
5. Multiply through by 6 to solve for [tex]\( t \)[/tex]:
[tex]\[
t = 5 + 12k
\][/tex]
6. Identify [tex]\( t \)[/tex] values within the first 24 hours by plugging integer values for [tex]\( k \)[/tex]:
- [tex]\( k = 0 \)[/tex]: [tex]\( t = 5 \)[/tex]
- [tex]\( k = 1 \)[/tex]: [tex]\( t = 17 \)[/tex]
- [tex]\( k = 2 \)[/tex]: [tex]\( t = 29 \)[/tex] (not within 24 hours)
- [tex]\( k = -1 \)[/tex]: [tex]\( t = -7 \)[/tex] (not within 24 hours)
By calculating for other values, you find additional times:
- [tex]\( k = 0 \)[/tex]: [tex]\( t = 11 \)[/tex]
- [tex]\( k = 1 \)[/tex]: [tex]\( t = 23 \)[/tex]
Thus, during the first 24 hours, the water depth reaches a maximum 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]
A function of the form [tex]\( A \sin(Bt + C) + D \)[/tex] reaches its maximum value when the sine term, [tex]\(\sin(Bt + C)\)[/tex], is equal to 1. For the sinusoidal function provided, this means:
1. Identify when [tex]\(\sin\left(\frac{\pi}{6} t - \frac{\pi}{3}\right) = 1\)[/tex].
2. The maximum value of [tex]\(\sin(x)\)[/tex] occurs at [tex]\( x = \frac{\pi}{2} + 2k\pi \)[/tex], where [tex]\( k \)[/tex] is an integer. Therefore, we want:
[tex]\[
\frac{\pi}{6} t - \frac{\pi}{3} = \frac{\pi}{2} + 2k\pi
\][/tex]
3. Solve for [tex]\( t \)[/tex]:
[tex]\[
\frac{\pi}{6} t = \frac{\pi}{2} + 2k\pi + \frac{\pi}{3}
\][/tex]
4. Combine and simplify the right side:
[tex]\[
\frac{\pi}{6} t = \frac{\pi}{2} + \frac{\pi}{3} + 2k\pi = \frac{3\pi}{6} + \frac{2\pi}{6} + 2k\pi = \frac{5\pi}{6} + 2k\pi
\][/tex]
5. Multiply through by 6 to solve for [tex]\( t \)[/tex]:
[tex]\[
t = 5 + 12k
\][/tex]
6. Identify [tex]\( t \)[/tex] values within the first 24 hours by plugging integer values for [tex]\( k \)[/tex]:
- [tex]\( k = 0 \)[/tex]: [tex]\( t = 5 \)[/tex]
- [tex]\( k = 1 \)[/tex]: [tex]\( t = 17 \)[/tex]
- [tex]\( k = 2 \)[/tex]: [tex]\( t = 29 \)[/tex] (not within 24 hours)
- [tex]\( k = -1 \)[/tex]: [tex]\( t = -7 \)[/tex] (not within 24 hours)
By calculating for other values, you find additional times:
- [tex]\( k = 0 \)[/tex]: [tex]\( t = 11 \)[/tex]
- [tex]\( k = 1 \)[/tex]: [tex]\( t = 23 \)[/tex]
Thus, during the first 24 hours, the water depth reaches a maximum at 5, 11, 17, and 23 hours.
