3. At the end of a dock, high tide of 14 m is recorded at 9:00 a.m. Low tide of 6 m is recorded at 3:00 p.m. A sinusoidal function can model the water depth versus time.

a) Construct a model for the water depth using a cosine function, where time is measured in hours past high tide.
b) Construct a model for the water depth using a sine function, where time is measured in hours past high tide.
c) Construct a model for the water depth using a sine function, where time is measured in hours past low tide.
d) Construct a model for the water depth using a cosine function, where time is measured in hours past low tide.
e) Compare your models. Which is the simplest representation if time is referenced to high tide? low tide? Explain why there is a difference.

Please help, if possible link images of the graphs. Thank you so much.

I suspect they are expecting you to say the cosine model is simpler, since then there is no shift needed: we start at the max or min.

Then all you need is

y = ±4cos(π/6 x) + 10

I got the rest, can you help me with part e)? I'm not sure what it is asking...

e) Compare your models. Which is the simplest representation if time is referenced to high tide? low tide? Explain why there is a difference.

However, sin(x)=0 at x=0 (high tide), and we want y to be at the center line 3 hours before high tide. So, shifting left 3 hours, we get

y = 4sin(π/6 (x+3)) + 10

Can you explain how you got that part?

sure. at t=0, we are at the maximum value. But sin(0) = 0 and sin(t) is a maximum 1/4 period later. Go to the website and play around with different shift values. If you replace 3 with 0, you will see that the graph starts at y=10, but we want to shift it so that it starts at y=14.

Between low tide and high tide, the width of a beach changes by −17 feet per hour. Write and evaluate an expression to show how much the width of the beach changes in 3 hours.

I need help?

To construct models for the water depth using sine and cosine functions, we need to first understand the properties and behavior of sine and cosine waves.

Both sine and cosine functions are periodic and oscillate between -1 and 1. The period of a sine or cosine function is the length of one complete cycle. In the context of tides, the period represents the time it takes for the water to go from high tide to high tide or from low tide to low tide.

In this problem, we are given that the high tide of 14 m is recorded at 9:00 a.m and the low tide of 6 m is recorded at 3:00 p.m. Based on these measurements, we can determine the period of the tide cycle.

a) To construct a model using a cosine function, where time is measured in hours past high tide, we need to find the time it takes for the water to go from high tide to high tide. From 9:00 a.m to 3:00 p.m is a total of 6 hours, which is the period of the tide cycle.

The general equation for a cosine function is: A * cos(B * (x - C)) + D, where A represents the amplitude, B represents the frequency (or 2π/period), C represents a horizontal shift (phase shift), and D represents a vertical shift.

For our model, the amplitude A is half the difference between high tide and low tide: (14 - 6) / 2 = 4 m. The frequency B is 2π divided by the period, which is 2π / 6 = π/3. The horizontal shift C is 0, since we're measuring time past high tide. And the vertical shift D is the average of high tide and low tide: (14 + 6) / 2 = 10 m.

Therefore, the model for the water depth using a cosine function would be: f(x) = 4 * cos((π/3) * x) + 10, where x represents the time in hours past high tide.

b) To construct a model using a sine function, where time is measured in hours past high tide, we can use the same period of 6 hours. The general equation for a sine function is similar to the cosine function: A * sin(B * (x - C)) + D.

Using the same values for amplitude A, frequency B, horizontal shift C, and vertical shift D as in part a, the model for the water depth using a sine function would be: f(x) = 4 * sin((π/3) * x) + 10.

c) To construct a model using a sine function, where time is measured in hours past low tide, we need to account for the fact that low tide occurs 6 hours after high tide. The period of the tide cycle remains 6 hours, but the horizontal shift C is now 6 to reflect the time past low tide.

Using the same values for A, B, and D as in parts a and b, the model for the water depth using a sine function, measured in hours past low tide, would be: f(x) = 4 * sin((π/3) * (x - 6)) + 10.

d) To construct a model using a cosine function, where time is measured in hours past low tide, we can use a similar approach as in part c. The period remains 6 hours, but the horizontal shift C is now 6.

Using the same values for A, B, and D as in parts a, b, and c, the model for the water depth using a cosine function, measured in hours past low tide, would be: f(x) = 4 * cos((π/3) * (x - 6)) + 10.

e) Comparing the models, we can see that for both time referenced to high tide and low tide, the sine and cosine functions are essentially the same, with only a horizontal shift of 6 hours. Therefore, both models are equally simple in terms of their representation.

It's worth noting that while we derived specific models based on the given data, in practice, the actual tide patterns can be more complex and may require additional terms or adjustments to fully capture the behavior. The models provided here serve as a simplified representation.

Here are the graphs for each of the functions mentioned above:
a) Model using a cosine function (time past high tide): ![Cosine Model](https://i.imgur.com/yBZp0np.png)
b) Model using a sine function (time past high tide): ![Sine Model](https://i.imgur.com/7Z5yWyd.png)
c) Model using a sine function (time past low tide): ![Sine Model (low tide)](https://i.imgur.com/bZW6wLj.png)
d) Model using a cosine function (time past low tide): ![Cosine Model (low tide)](https://i.imgur.com/azHu5L4.png)

high tide = 14m

low tide = 6m
so, the center line is (14+6)/2 = 10m, and the amplitude is (14-6)/2 = 4m

The 1/2 period (max to min) is 6 hours, so the period is 12 hours. SO, we can start with

y = 4sin(?/6 x) + 10

However, sin(x)=0 at x=0 (high tide), and we want y to be at the center line 3 hours before high tide. So, shifting left 3 hours, we get

y = 4sin(?/6 (x+3)) + 10

check it out here:

http://www.wolframalpha.com/input/?i=plot+y%3D4sin(%CF%80%2F6+(x%2B3))+%2B+10,+y%3D10

Note that the low tide is at x=6, or 6 hours after high tide (3 pm), as required.

I'm sure you can now resolve the other questions. Use the web site to test your formulas.