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Problem-solving Pack

Ecolint Campus des NationsMathematics
Year 11 · 11.9 Trigonometric Modelling

Problem-solving Pack

Name: _________________________________
Date: _________________ Class: ___________

These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.

1Problem 1 of 12
Tide model. The depth dd (m) of water in a harbour follows d(t)=3sin⁡ ⁣(πt6)+5d(t) = 3\sin\!\left(\dfrac{\pi t}{6}\right) + 5, where tt is hours after midnight.

(a) State the amplitude, period, and mean depth.
(b) Find the maximum and minimum depths.
(c) Find the first time after midnight at which the depth is exactly 7 m, to 3 s.f.

Working space

2Problem 2 of 12
Ferris wheel. A Ferris wheel of radius 15 m has its centre 18 m above ground. It rotates anti-clockwise with period 4 minutes. A capsule starts at the lowest point at t=0t = 0.

(a) Explain why h(t)=18−15cos⁡ ⁣(πt2)h(t) = 18 - 15\cos\!\left(\dfrac{\pi t}{2}\right) models its height.
(b) State the max and min heights.
(c) Find the first time the capsule is 25 m above ground, to 3 s.f.
(d) State, without further calculation, the total time per revolution that the capsule is above 25 m.

Working space

3Problem 3 of 12
Periodic features. For y=Asin⁡(Bx)+Cy = A\sin(Bx) + C with A=3A = 3, B=π6B = \dfrac{\pi}{6}, C=4C = 4:

(a) State the amplitude.
(b) State the period.
(c) State the maximum and minimum values of yy.
(d) Sketch yy for 0≤x≤120 \leq x \leq 12, marking the yy-intercept and the first maximum.

Working space

4Problem 4 of 12
Daylight in a city. Daylight varies from 9 h on 21 December (t=0t = 0, where tt is in months) to 15 h on 21 June (t=6t = 6).

(a) State the amplitude and the mean.
(b) Write a model L(t)=−Acos⁡ ⁣(πt6)+DL(t) = -A\cos\!\left(\dfrac{\pi t}{6}\right) + D.
(c) Predict the daylight on 21 March (t=3t = 3).
(d) Find tt where the model first gives L=13L = 13 h, to 3 s.f.

Working space

5Problem 5 of 12
Pendulum. A pendulum swings horizontally and its position from rest follows x(t)=10cos⁡(πt)x(t) = 10\cos(\pi t), where xx is cm and tt is seconds.

(a) State the amplitude and period.
(b) Find x(0)x(0), x(0.5)x(0.5), x(1)x(1).
(c) Find the times in [0,2][0, 2] s at which x=0x = 0.
(d) Find the maximum speed (in cm/s) by symbolic differentiation, v(t)=x′(t)v(t) = x'(t).

Working space

6Problem 6 of 12
Convert sin ↔ cos. Rewrite each function using a single sine OR cosine and a phase shift.

(a) y=sin⁡xy = \sin x as a cosine
(b) y=cos⁡xy = \cos x as a sine
(c) y=−cos⁡xy = -\cos x as a cosine with a phase shift

Working space

7Problem 7 of 12
Fitting from a table. Data observed:

| tt (s) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| yy | 5 | 9 | 5 | 1 | 5 | 9 |

(a) State the period.
(b) State the amplitude and mean.
(c) Write a model y(t)=Asin⁡(Bt)+Cy(t) = A\sin(Bt) + C given that y(0)y(0) is at the mean and rising.

Working space

8Problem 8 of 12
Daylight + threshold. Using L(t)=−3cos⁡(πt/6)+12L(t) = -3\cos(\pi t / 6) + 12 (h, tt in months from 21 Dec):

(a) For how many months per year is L>13L > 13 h?
(b) State the start and end times of this interval, to 3 s.f.
(c) Sketch LL for 0≤t≤120 \leq t \leq 12 marking L=13L = 13 on the yy-axis.

Working space

9Problem 9 of 12
Periodic vs non-periodic. Decide whether each function is periodic. Justify, and (where periodic) state the period.

(a) y=sin⁡x+cos⁡xy = \sin x + \cos x
(b) y=sin⁡x+xy = \sin x + x
(c) y=sin⁡(2x)+sin⁡(3x)y = \sin(2x) + \sin(3x) [EXT]
(d) y=sin⁡x⋅cos⁡xy = \sin x \cdot \cos x

Working space

10Problem 10 of 12
Modelling — sound wave. A sound wave is modelled by P(t)=0.5sin⁡(2π⋅440⋅t)P(t) = 0.5\sin(2\pi \cdot 440 \cdot t), where PP is pressure in Pa and tt in seconds.

(a) State the amplitude and frequency.
(b) Find the period.
(c) Find P(0)P(0), P(1/1760)P(1/1760).
(d) Comment on what the frequency 440 Hz represents musically.

Working space

11Problem 11 of 12
Inverse modelling. A tidal model gives d(t)=4sin⁡ ⁣(πt6)+7d(t) = 4\sin\!\left(\dfrac{\pi t}{6}\right) + 7 where tt is hours since the previous high tide (so dd peaks at t=3t = 3).

(a) State the amplitude, period, and average depth.
(b) Find the depth at t=0t = 0.
(c) Find the first t>0t > 0 at which the depth is again 7 m.
(d) Find the duration of one "low-tide window" defined as d<5d < 5.

Working space

12Problem 12 of 12
Investigation — fit & predict. Population of foxes in a region is observed monthly:

| tt | 1 | 4 | 7 | 10 |
|---|---|---|---|---|
| PP | 410 | 240 | 410 | 580 |

(a) Argue that a sinusoidal model is reasonable.
(b) Estimate amplitude, mean, and period from the data.
(c) Fit a model P(t)=Asin⁡(B(t−C))+DP(t) = A\sin(B(t - C)) + D.
(d) Use the model to predict the population at t=12t = 12.

Working space