the water depth in a harbor rises and falls over time. the function (f(t)=4.1sin(\frac{pi}{6}t…

Explanation:

Step1: Recall sine - function maximum

The maximum value of the sine function $y = A\sin(Bx - C)+D$ occurs when $\sin(Bx - C)=1$. For the function $f(t)=4.1\sin(\frac{\pi}{6}t-\frac{\pi}{3}) + 19.7$, we set $\sin(\frac{\pi}{6}t-\frac{\pi}{3}) = 1$.

Step2: Solve for $t$ when $\sin(\theta)=1$

We know that $\sin\theta = 1$ when $\theta=\frac{\pi}{2}+2k\pi$, $k\in\mathbb{Z}$. So, we set $\frac{\pi}{6}t-\frac{\pi}{3}=\frac{\pi}{2}+2k\pi$. First, simplify the equation: [ \begin{align*} \frac{\pi}{6}t-\frac{\pi}{3}&=\frac{\pi}{2}+2k\pi\ \frac{\pi}{6}t&=\frac{\pi}{2}+\frac{\pi}{3}+2k\pi\ \frac{\pi}{6}t&=\frac{3\pi + 2\pi}{6}+2k\pi\ \frac{\pi}{6}t&=\frac{5\pi}{6}+2k\pi \end{align*} ] Then, multiply both sides by $\frac{6}{\pi}$ to solve for $t$: [ \begin{align*} t&=5 + 12k \end{align*} ]

Step3: Find values of $t$ in the first 24 - hour period

We want to find values of $t$ such that $0\leq t\leq24$. When $k = 0$, $t=5$. When $k = 1$, $t=5 + 12=17$. When $k = 2$, $t=5+24>24$.

Answer:

at 5 and 17 hours