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Math 30-1 · Worksheets

Sinusoidal models · Set B

Ten questions of mixed difficulty, covering Sinusoidal models. Print it, or work through it on screen — the answer key starts on its own page.

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Sinusoidal models · Set B

Math 30-1 · maddyhelps.com

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  1. A Ferris wheel has a radius of 15 m, its centre is 17 m up, one rotation takes 60 s, and the rider boards at the bottom at t = 0. Which model is correct?

    1. a) h(t) = −30 cos(πt/30) + 15
    2. b) h(t) = −15 cos(πt/30) + 17
    3. c) h(t) = −15 cos(πt/60) + 17
    4. d) h(t) = 15 cos(πt/30) + 17

  2. In a sinusoidal model of a rider's height on a Ferris wheel, what does the amplitude represent?

    1. a) the diameter of the wheel
    2. b) the height of the wheel's centre
    3. c) the radius of the wheel
    4. d) the time for one rotation

  3. Which equation has an amplitude of 3, a period of π and a midline of y = −2?

    1. a) y = −2 sin 3x + π
    2. b) y = 3 sin 2x − 2
    3. c) y = 2 sin 3x − 3
    4. d) y = 3 sin(½x) − 2

  4. What is the horizontal shift of y = 2 sin(3x − π) + 1?

    1. a) π to the right
    2. b) π/3 to the right
    3. c) 3π to the right
    4. d) π/3 to the left

  5. The tide is modelled by h(t) = 2 sin(πt/6) + 5, in metres. How high is high tide?

    1. a) 7 m
    2. b) 12 m
    3. c) 2 m
    4. d) 5 m

  6. A Ferris wheel has a diameter of 20 m and its centre is 12 m above the ground. What are a rider's maximum and minimum heights?

    1. a) 32 m and 12 m
    2. b) 12 m and 2 m
    3. c) 22 m and 2 m
    4. d) 20 m and 0 m

  7. A sinusoidal graph has a maximum of 9 and a minimum of 1. What is its amplitude?

    1. a) 5
    2. b) 8
    3. c) 9
    4. d) 4

  8. A rider's height is h(t) = −10 cos(πt/20) + 12, in metres after t seconds. How high is the rider at t = 0?

    1. a) 12 m
    2. b) 2 m
    3. c) −10 m
    4. d) 22 m

  9. A sinusoidal graph has a maximum at (2, 7) and the next minimum at (8, 1). Which equation fits?

    1. a) y = 3 cos[π/6 (x − 2)] + 4
    2. b) y = 3 cos[π/12 (x − 2)] + 4
    3. c) y = 3 cos[π/6 (x + 2)] + 4
    4. d) y = 6 cos[π/6 (x − 2)] + 4

  10. In a sinusoidal model of a Ferris wheel, the period represents:

    1. a) the time for one full rotation
    2. b) the height of the centre
    3. c) the radius of the wheel
    4. d) the highest point a rider reaches

Answer key · Sinusoidal models · Set B

Math 30-1 · maddyhelps.com

  1. b) h(t) = −15 cos(πt/30) + 17 — a = 15 (radius), d = 17 (centre), b = 2π ÷ 60 = π/30. Starting at the bottom means starting at a minimum, which a negative cosine does.
  2. c) the radius of the wheel — The rider goes one radius above and one radius below the centre. The centre's height is the midline, and one rotation is the period.
  3. b) y = 3 sin 2x − 2 — Amplitude 3 means a = 3. Period π means b = 2π ÷ π = 2. Midline −2 means d = −2. b = ½ would give a period of 4π.
  4. b) π/3 to the right — Factor the 3 out first: 3(x − π/3). The shift is π/3 right. Reading π straight from the bracket ignores the factoring.
  5. a) 7 m — High tide is the midline plus the amplitude: 5 + 2 = 7 m. Low tide would be 5 − 2 = 3 m.
  6. c) 22 m and 2 m — The radius is 10 m, so the rider goes from 12 − 10 = 2 m up to 12 + 10 = 22 m. Using the diameter instead of the radius is the usual slip.
  7. d) 4 — Amplitude is half the distance from minimum to maximum: (9 − 1) ÷ 2 = 4. 8 is the whole height.
  8. b) 2 m — cos 0 = 1, so h(0) = −10 + 12 = 2 m. The negative cosine starts at its minimum — the rider boards at the bottom.
  9. a) y = 3 cos[π/6 (x − 2)] + 4 — Amplitude (7 − 1) ÷ 2 = 3, midline 4, period 12 so b = π/6. Cosine starts at a maximum, and the maximum is at x = 2, so the shift is 2 right: (x − 2).
  10. a) the time for one full rotation — One period is one complete cycle — for a wheel, one trip all the way around.