To model the tide using a sine function based on the given information, we need to determine key components of the sine equation and fit it to the pattern of the graph. The standard sine function is of the form:

$y(t) = A \sin(B(t - C)) + D$

Where:

• $A$ is the amplitude (half the distance between the maximum and minimum values),
• $B$ is related to the period (how long it takes for the cycle to repeat),
• $C$ is the horizontal shift (the time at which the sine function reaches its maximum),
• $D$ is the vertical shift (the average value or the middle of the oscillation).

Step 1: Determine the vertical shift $D$

From the problem, the average depth of water is 6 feet. This tells us that the sine wave oscillates around 6 feet. Therefore, the vertical shift $D = 6$.

$D = 6$

Step 2: Determine the amplitude $A$

The tide varies 2 feet in either direction from the average (6 feet), meaning the maximum depth is 8 feet and the minimum depth is 4 feet. The amplitude is the distance from the middle (average) to either extreme, which is 2 feet.

$A = 2$

Step 3: Determine the period and calculate $B$

The tide goes from low to high every 6 hours, meaning it takes a total of 12 hours for a full cycle (from high tide back to high tide). The period of a sine function is the time it takes to complete one full cycle. The general formula for the period of a sine function is:

$\text{Period} = \frac{2\pi}{B}$

Since the period is 12 hours:

$12 = \frac{2\pi}{B} \implies B = \frac{2\pi}{12} = \frac{\pi}{6}$

Step 4: Determine the phase shift $C$

High tide occurs at 3 a.m. In a standard sine function, the maximum occurs at $t = 0$, but in this case, the maximum occurs at $t = 3$. This means the graph is shifted 3 hours to the right. Thus, $C = 3$.

Step 5: Write the sine function

Now that we have all the components:

• Amplitude $A = 2$,
• Vertical shift $D = 6$,
• Period coefficient $B = \frac{\pi}{6}$,
• Phase shift $C = 3$.

The sine function modeling the tide is:

$y(t) = 2 \sin\left(\frac{\pi}{6}(t - 3)\right) + 6$

Final Answer:

The sine function that models the tide is:

$y(t) = 2 \sin\left(\frac{\pi}{6}(t - 3)\right) + 6$

Would you like more details or have any questions?

Further Questions:

1. How would the equation change if the high tide were at midnight instead of 3 a.m.?
2. Can this sine function be used to predict future tides? How accurate is it?
3. What is the depth of the water at 9 a.m. using this model?
4. How would you adjust the function if the tide varied by 3 feet instead of 2 feet?
5. What would the graph look like if the period changed from 12 hours to 24 hours?

Tip:

The phase shift ($C$) is always determined by where the sine wave reaches its peak or zero-crossing, so identifying this point from the graph is crucial for creating an accurate model.