Question

the water depth in a harbor rises and falls over time. the function ( f(t)=4.1sinleft(\frac{pi}{6}t - \frac{pi}{3}
ight)+19.7 ) models the water depth, in feet, after ( t ) hours. during the first 24 hours, at what times does the water depth reach a maximum? at 5 and 17 hours at 11 and 23 hours at 2, 8, 14, and 20 hours at 5, 11, 17, and 23 hours

Explanation:

Step1: Find the general solution for maximum

The maximum of \(y = A\sin(Bt - C)+D\) occurs when \(Bt - C=\frac{\pi}{2}+ 2k\pi\), \(k\in\mathbb{Z}\).
For \(f(t)=4.1\sin(\frac{\pi}{6}t-\frac{\pi}{3}) + 19.7\), set \(\frac{\pi}{6}t-\frac{\pi}{3}=\frac{\pi}{2}+2k\pi\).

Step2: Solve for \(t\)

First, simplify the equation \(\frac{\pi}{6}t-\frac{\pi}{3}=\frac{\pi}{2}+2k\pi\).
Add \(\frac{\pi}{3}\) to both sides: \(\frac{\pi}{6}t=\frac{\pi}{2}+\frac{\pi}{3}+2k\pi=\frac{3\pi + 2\pi}{6}+2k\pi=\frac{5\pi}{6}+2k\pi\).
Multiply both sides by \(\frac{6}{\pi}\): \(t = 5+12k\).

Step3: Find \(t\) values in \(0\leq t\leq24\)

When \(k = 0\), \(t=5\).
When \(k = 1\), \(t=5 + 12=17\).
When \(k = 2\), \(t=5+24 = 29>24\) (rejected).
When \(k=- 1\), \(t=5-12=-7<0\) (rejected).

Answer:

at 5 and 17 hours