I am currently working on a problem related to water depth measurements and I am seeking some help with it. The problem is as follows:
Water depth at a dock is measured during the first 16 hours of a day, as shown in Figure 1. The measurements showed that the greatest depth of 178 centimeters occurred at midnight and 1:00 pm, and the depth was 0 centimeters for five hours. When the water depth is greater than 0 centimeters, it can be described as $y = A \cos(kx) + d$, where $y$ is the water depth in centimeters, and $x$ is the time in hours after midnight, as shown in Figure 2.
a) Determine the value of the constant $k$. Provide only the answer. b) Determine the values of the constants $A$ and $d$.
I would appreciate any help or guidance on how to approach this problem. Thank you in advance!
For part a, note that $\cos(kx)$ is a periodic function. It has a maximum at $x=0$. Then the next maximum is at $kx=2\pi$. The $x$ for the second maximum is 13 hours.
For part b, once you figure out $k$ in the previous part, you need to write two equations to find two variables ($A$ and $d$). The first one is at $x=0$, $$A+d=178$$ For the second equation, you need to find out when the zero depth starts. Let's call this time $t_0$. Then $$A\cos(kt_0)+d=0$$ Solve the system of equations.