High and low tide at a dock

The question

Name: ______________________   Date: ____________   High and low tide at a dock (6 points)

Graph

Depth of water at a dock

The depth of water at the end of a dock changes with the tide. On one day, high tide, with a depth of 3.1 meters, occurs at 2 a.m., and the next low tide, with a depth of 0.7 meters, occurs at 8 a.m. The sinusoidal function $W$ models the depth of the water, in meters, $t$ hours after midnight. In this model, high tides are 12 hours apart. The graph of $W$ for $2 \le t \le 14$ is shown with its axes unscaled. The points $P$, $Q$, $R$, $S$ and $T$ are on the graph: $P$ and $T$ are highest points, $R$ is a lowest point, and $Q$ and $S$ are on the midline.

1. (A) Find the coordinates of the points $P$, $Q$, $R$, $S$ and $T$. [2 points]

2. (B) The function $W$ is given by $W(t)$$= a\cos(b(t + c)) + d$, where $a$, $b$, $c$ and $d$ are constants. Find values of $a$, $b$, $c$ and $d$. [2 points]

3. (C(i)) Is $W$ increasing or decreasing on the interval $(8, 11)$? Give a reason for your answer in the context of the problem. [1 point]

4. (C(ii)) Is the rate of change of $W$ increasing or decreasing on the interval $(8, 11)$? Explain your reasoning. [1 point]