Answer :
To find when the water depth in the harbor 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]
This function is sine-based, and the maximum value of the sine function, [tex]\(\sin(x)\)[/tex], is when it equals 1. So we need the expression inside the sine function to result in a value where the sine function equals 1.
Step-by-step Solution:
1. Identify the Condition for Maximum:
[tex]\[
\sin\left(\frac{\pi}{6} t - \frac{\pi}{3}\right) = 1
\][/tex]
This occurs when:
[tex]\[
\frac{\pi}{6} t - \frac{\pi}{3} = \frac{\pi}{2} + 2n\pi
\][/tex]
for any integer [tex]\(n\)[/tex].
2. Solve for [tex]\(t\)[/tex]:
- Add [tex]\(\frac{\pi}{3}\)[/tex] to both sides:
[tex]\[
\frac{\pi}{6} t = \frac{\pi}{2} + \frac{\pi}{3} + 2n\pi
\][/tex]
- Simplify the right side:
[tex]\[
= \frac{\pi}{2} + \frac{2\pi}{6} + 2n\pi = \frac{3\pi + 2\pi}{6} + 2n\pi = \frac{5\pi}{6} + 2n\pi
\][/tex]
- Multiply both sides by [tex]\(\frac{6}{\pi}\)[/tex] to solve for [tex]\(t\)[/tex]:
[tex]\[
t = \frac{6}{\pi} \left(\frac{5\pi}{6} + 2n\pi\right) = 5 + 12n
\][/tex]
3. Find Appropriate Values for [tex]\(t\)[/tex] within the First 24 Hours:
We need [tex]\(t\)[/tex] to satisfy [tex]\(0 \leq t < 24\)[/tex]:
- 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].
- For [tex]\(n = 2\)[/tex], [tex]\(t = 5 + 12 \times 2 = 29\)[/tex] (This value is outside the range of the first 24 hours).
This pattern shows that the water depth reaches a maximum at:
- 5 hours
- 17 hours
Therefore, during the first 24 hours, the water depth reaches maximum at 5 and 17 hours.
[tex]\[ f(t) = 4.1 \sin\left(\frac{\pi}{6} t - \frac{\pi}{3}\right) + 19.7 \][/tex]
This function is sine-based, and the maximum value of the sine function, [tex]\(\sin(x)\)[/tex], is when it equals 1. So we need the expression inside the sine function to result in a value where the sine function equals 1.
Step-by-step Solution:
1. Identify the Condition for Maximum:
[tex]\[
\sin\left(\frac{\pi}{6} t - \frac{\pi}{3}\right) = 1
\][/tex]
This occurs when:
[tex]\[
\frac{\pi}{6} t - \frac{\pi}{3} = \frac{\pi}{2} + 2n\pi
\][/tex]
for any integer [tex]\(n\)[/tex].
2. Solve for [tex]\(t\)[/tex]:
- Add [tex]\(\frac{\pi}{3}\)[/tex] to both sides:
[tex]\[
\frac{\pi}{6} t = \frac{\pi}{2} + \frac{\pi}{3} + 2n\pi
\][/tex]
- Simplify the right side:
[tex]\[
= \frac{\pi}{2} + \frac{2\pi}{6} + 2n\pi = \frac{3\pi + 2\pi}{6} + 2n\pi = \frac{5\pi}{6} + 2n\pi
\][/tex]
- Multiply both sides by [tex]\(\frac{6}{\pi}\)[/tex] to solve for [tex]\(t\)[/tex]:
[tex]\[
t = \frac{6}{\pi} \left(\frac{5\pi}{6} + 2n\pi\right) = 5 + 12n
\][/tex]
3. Find Appropriate Values for [tex]\(t\)[/tex] within the First 24 Hours:
We need [tex]\(t\)[/tex] to satisfy [tex]\(0 \leq t < 24\)[/tex]:
- 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].
- For [tex]\(n = 2\)[/tex], [tex]\(t = 5 + 12 \times 2 = 29\)[/tex] (This value is outside the range of the first 24 hours).
This pattern shows that the water depth reaches a maximum at:
- 5 hours
- 17 hours
Therefore, during the first 24 hours, the water depth reaches maximum at 5 and 17 hours.
