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AI SL · 12.7

Mathematical Models

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12.7.2

Sinusoidal models: Ferris wheels, tides and daylight

12.7.2

Sinusoidal models: Ferris wheels, tides and daylight

Target: I can build a model \(a\sin(bt)+d\) or \(a\cos(bt)+d\) from the maximum, the minimum, the period and where the motion starts. (+3 more)

By the end of this section

Key points

Building the model

  • Amplitude \(a=\dfrac{\max-\min}{2}\). Mean (central line) \(d=\dfrac{\max+\min}{2}\). \(b=\dfrac{360}{\text{period}}\).
  • Starts at the mean and rising: \(h=a\sin(bt)+d\).
  • Starts at the maximum: \(h=a\cos(bt)+d\).
  • Starts at the minimum (a Ferris wheel at the bottom): \(h=-a\cos(bt)+d\), with a positive. Check: at t = 0, \(h=-a+d\) is the minimum.
  • Set the calculator to degrees. Graph the model and check the maximum, the minimum and the length of one period.

Using the model

  • A value at a time: put t in. Keep b unrounded, or use the stored value.
  • A level is reached twice in each cycle (once going up, once coming down). From the calculator: graph the model and the line, and find the intersections. Or solve model = level for t inside one cycle.
  • Time above a level in one cycle = the later crossing time − the earlier crossing time.
  • Many solutions exist for \(t\ge0\). Use the domain or the cycle the question asks about.

What the model assumes

  • Every cycle is exactly the same. A Ferris wheel turns at a constant speed, so the model suits it well. A real tide, a real temperature or the daylight hours change a little from one cycle to the next.
  • Say what a, b and d mean in the context: a is the distance above and below the mean, the period is the time for one cycle, d is the middle value.
  • A model of a bounded quantity is only reasonable where the quantity makes sense.

On the calculator

Graphing a model to check it

  • Check the angle mode is degrees.
  • Graph the model. Check the maximum, the minimum and the length of one period against the question.
  • Set the window to show at least one full period.

Screenshot and video to come

Finding the times a level is reached

  • Graph the model and the horizontal line at the level. Find the intersections.
  • Or write the model equal to the level and solve for t, with t limited to one cycle.
  • Read both times. The time above the level is the difference.

Screenshot and video to come

Explore: a sin(bt) + d, the tide and the Ferris wheel

These are depth readings from a harbour. Before you move anything, work out the mean d and the amplitude a from the highest and lowest readings. Then drag the orange points on the curve, or use the sliders, until the curve passes close to the readings. The teal dots are the readings that are already close.

Range (min to max)
Readings within 0.2 m

Move a slider.

  1. Which point moves the whole curve up and down? Which one changes the height of the peaks? How can you tell d and a apart on the graph?
  2. Change the period T. What happens to b? Check that b × T is always 360.
  3. Find a model that fits, then change d by 1. What happens to the curve? Which readings are now wrong?
  4. Why is the highest depth d + a and the lowest d − a?
  5. On the Ferris wheel, the rider gets on at the bottom. Why is the model −a cos(bt) + d and not a sin(bt) + d? Where is the rider after half a turn?

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
A Ferris wheel from the bottom

A Ferris wheel has a diameter of 24 m and its lowest point is 2 m above the ground. It turns once every 8 minutes. A rider gets on at the lowest point at t = 0, where t is in minutes. The height of the rider is \(h=a\cos(bt)+d\). (a) Find a, b and d. (b) Find the height after 3 minutes. (c) Find the first time the rider is 20 m above the ground.

246810121416510152025t (minutes)h (m)h
Working
  1. The extremes

    Maximum 2 + 24 = 26, minimum 2.

    Diameter 24 means the height changes by 24 from bottom to top.

  2. a and d

    d = (26 + 2) ÷ 2 = 14, |a| = (26 − 2) ÷ 2 = 12. The rider starts at the minimum, so a = −12.

    Cosine starts at its maximum, so the minus sign turns it upside down.

  3. b

    \(b=360\div8=45\)

    Degrees.

  4. The model

    \(h=-12\cos(45t)+14\)

    Check by graphing: at t = 0, h = 2.

  5. Part (b)

    \(h=-12\cos(45\times3)+14=22.5\)

    After 3 minutes the rider is 22.5 m up.

  6. Part (c)

    Solve for t: h = 20 gives t = 2.67 minutes

    First crossing: choose the smallest positive solution.

Video walkthrough coming
How long can the boat enter?

The depth of water at a harbour entrance is 5.6 m at high tide and 1.8 m at low tide. High tides are 12.4 hours apart. At t = 0, where t is in hours, the depth is at its mean and rising. The depth is \(D=a\sin(bt)+d\). (a) Find a, b and d. (b) A boat needs a depth of at least 4.5 m to enter. Find the total time in the first 12.4 hours for which the boat can enter.

Working
  1. a and d

    a = (5.6 − 1.8) ÷ 2 = 1.9, d = (5.6 + 1.8) ÷ 2 = 3.7

    Mean start, rising: sine.

  2. b

    \(b=360\div12.4=29.0\)

    Degrees.

  3. The model

    \(D=1.9\sin(29.0t)+3.7\)

    Graph it: maximum 5.6, minimum 1.8.

  4. Crossings

    Solve for t: D = 4.5 gives t = 0.858 and t = 5.34

    Two times in one cycle: going up, then coming down.

  5. Time above

    5.34 − 0.858 = 4.48 hours

    Between the crossings the depth is above 4.5 m.

Video walkthrough coming
Hours of daylight

In a northern city the length of the day is 15.4 hours at its longest and 8.6 hours at its shortest. The pattern repeats every 12 months. Let t be the months after the spring equinox, when the day is 12 hours long and getting longer. The length of the day is \(L=a\sin(bt)+d\). (a) Find a, b and d. (b) Find L when t = 5. (c) Say what a and d tell you about the days. Give one reason the model is not exact.

Working
  1. a and d

    a = (15.4 − 8.6) ÷ 2 = 3.4, d = (15.4 + 8.6) ÷ 2 = 12

    At the equinox the day is 12 hours: the mean.

  2. b

    \(b=360\div12=30\)

    The period is 12 months.

  3. Part (b)

    \(L=3.4\sin(30\times5)+12=13.7\)

    About 13.7 hours.

  4. Meaning

    a = 3.4: the day is up to 3.4 hours longer or shorter than 12 hours. d = 12: the average day.

    Say what the numbers do in the context.

  5. Limits

    The model repeats exactly every year and the real year is 365.25 days, not 12 equal months.

    Real data is not perfectly sinusoidal.

Video walkthrough coming