Math

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.

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  1. 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.

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  2. 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.

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  3. 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

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  4. 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?

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  5. 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.

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  6. 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?

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