Modelling periodic situations

← Previous5 of 5Next →
Worksheet with answersDownload PDF

What this lesson covers

  • The midline is the average of the maximum and minimum.
  • The amplitude is half their difference.
  • b = 2π ÷ period.
Key idea

Building the model

  • Use a cosine if it starts at a maximum.
  • Use −cos if it starts at a minimum.
  • Use sine if it starts on the midline, going up.
Worked example

A tide

  • High tide is 7 m at t = 0 and low tide is 3 m six hours later.
  1. midline = 5, amplitude = 2
  2. period 12, so b = π/6
  3. h = 2 cos(πt/6) + 5
Worked example

When is the water 6 m deep?

  • Solve with the model.
  1. 2 cos(πt/6) + 5 = 6
  2. cos(πt/6) = 1/2
  3. πt/6 = π/3 or 5π/3
  4. t = 2 and t = 10 hours

Common mistakes

  • Using the time from high to low as the period.
    That is half a period.
  • Giving only the first solution.
    Check for every time in the range asked.

Your turn

  • 1. A wheel has radius 10 m, centre 12 m up, and turns in 40 s, starting at the bottom. Write the height.
  • 2. When does it first reach 17 m?
Show answer

1. h = 12 − 10 cos(πt/20) 2. cos(πt/20) = −1/2, so t = 13.3 s

Summary

What to take away

  • Max and min give the midline and amplitude.
  • The period gives b.

More in AS 91575

Trig functions and radiansNotesTrig identities and proofsNotesCompound and double-angle formulaeNotesTrig equations and general solutionsNotesModelling periodic situationsNotes

Is your child on track?

Take the free placement test