Let's analyze the problem:

The depth of water at the boat dock follows a periodic pattern, with low tide at 8 feet and high tide at 14 feet. We are asked to model this depth using a cosine function of the form:

$y = A \cos(Bx + C) + D$

Step 1: Amplitude (A)

• The amplitude is the distance from the average depth to the extreme depth (either high tide or low tide).
• The average depth is the midpoint between the high and low tides: $\text{Average depth} = \frac{14 + 8}{2} = 11 \text{ feet}$
• The amplitude is the difference between the average and high tide: $A = 14 - 11 = 3$

Step 2: Vertical Shift (D)

• The vertical shift $D$ is the average depth, which we already calculated: $D = 11$

Step 3: Period and Frequency (B)

• The period of the function is the time it takes for one complete cycle of high tide and low tide. According to the graph, low tide occurs at 6 AM, and high tide at noon. Thus, the period is 12 hours.
• The period $T$ of a cosine function is related to $B$ by: $T = \frac{2\pi}{B}$ Since the period is 12 hours: $12 = \frac{2\pi}{B} \implies B = \frac{\pi}{6}$

Step 4: Phase Shift (C)

• To determine $C$, we need to analyze the phase shift. Low tide occurs at 6 AM (6 hours after midnight), so the cosine function should be shifted to match this.
  • The standard cosine function starts at its maximum when $x = 0$, but here, it reaches its minimum at $x = 6$.
  • A cosine function shifted by $\frac{\pi}{6}$ to the right will reach its minimum at $x = 6$: $C = -\frac{\pi}{2}$

Final Equation:

Now we can combine all the components into the equation:

$y = 3 \cos\left(\frac{\pi}{6}x - \frac{\pi}{2}\right) + 11$

This models the depth of the water in feet as a function of hours after midnight.

Would you like further details on any part of this solution, or do you have any questions?

Related Questions:

1. How is the amplitude determined in sinusoidal models of tides?
2. Why is the phase shift negative in this problem?
3. Can the sine function also model this situation? How would the equation differ?
4. What would the equation look like if the low tide occurred at 3 AM instead of 6 AM?
5. How does the vertical shift affect the graph of a cosine function?

Tip:

When working with periodic functions, always pay close attention to the period and phase shift, as they directly influence the shape and timing of the function's peaks and troughs.