Modelling periodic phenomena

Model real periodic situations — tides, temperature, daylight hours — using transformed trigonometric functions; fit amplitude, period and phase shift to data; use a model to find values and solve equations in context.

Worked examples

Reading amplitude, period and midline from a model

Straightforward

Problem

A tide is modelled by

H(t) = 2.8\sin\!\left(\dfrac{\pi t}{6}\right) + 4.2

, where

H

is height in metres and

t

is hours after midnight. State the amplitude, period and midline. Find the maximum and minimum heights.

Fitting a model to data

Moderate

Problem

The daily maximum temperature (°C) in a coastal town is recorded each month. The highest average is

32

°C in January (month

1

) and the lowest is

10

°C in July (month

7

). Fit a cosine model of the form

T(m) = a\cos(b(m - c)) + d

Using a model to solve for time

Challenging

Problem

Using the tidal model

H(t) = 3\sin\!\left(\dfrac{\pi t}{6}\right) + 4

, find all times in the first

24

hours when

H = 6.5

Practise

Q1·Straightforward

The height of the tide at a harbour is modelled by

H(t) = 3\sin\!\left(\dfrac{\pi t}{6}\right) + 4

, where

H

is the height in metres and

t

is time in hours after midnight.

What is the maximum height of the tide in metres?

Q2·Straightforward

The height of the tide is modelled by

H(t) = 3\sin\!\left(\dfrac{\pi t}{6}\right) + 4

, where

t

is in hours.

What is the period of the tide in hours?

Q3·Straightforward

The average monthly temperature (°C) in a city is modelled by

T(m) = 12\cos\!\left(\dfrac{2\pi m}{12}\right) + 18

, where

m

is the month number with

m = 0

representing the warmest month.

What is the minimum temperature in °C?

Q4·Straightforward

The depth of water (metres) at a pier is modelled by

D(t) = 2.5\sin\!\left(\dfrac{\pi t}{6} - \dfrac{\pi}{3}\right) + 5

, where

t

is time in hours.

State the amplitude of the depth in metres.

Q5·Moderate

A Ferris wheel has a maximum height of

30

m and a minimum height of

2

m above the ground. Riders board at the lowest point, and the wheel takes

4

minutes to complete one full revolution.

A model of the form

h(t) = -a\cos(bt) + d

is used, where

t

is time in minutes after boarding.

Find the value of

a

Q6·Moderate

The temperature

T

(°C) inside a greenhouse follows the model

T(t) = 8\sin\!\left(\dfrac{\pi(t-6)}{12}\right) + 22

, where

t

is hours after midnight,

0 \le t \le 24

.

At what time (hours after midnight) does the temperature first reach its daily maximum? Give an integer answer.

Q7·Moderate

The height of the tide is modelled by

H(t) = 1.8\cos\!\left(\dfrac{\pi t}{6}\right) + 3.2

, where

H

is in metres and

t

is hours after midnight.

Find the first time after midnight (to 2 decimal places) when the tide height equals

4

Q8·Moderate

The number of hours of daylight in a city in the northern hemisphere is modelled by

D(m) = 3\sin\!\left(\dfrac{\pi m}{6} - \dfrac{\pi}{2}\right) + 12

, where

m

is the month number (January

= 1

, December

= 12

).

Find the number of hours of daylight in April (

m = 4

Q9·Moderate

Using the daylight model

D(m) = 3\sin\!\left(\dfrac{\pi m}{6} - \dfrac{\pi}{2}\right) + 12

(northern hemisphere, January

= 1

), find the month number

m

in which daylight is longest.

Q10·Challenging

Tide measurements at a wharf give the following data:

$t$ (hours after midnight)	$0$	$3$	$6$	$9$	$12$
$H$ (metres)	$5$	$8$	$5$	$2$	$5$

A sine model

H(t) = a\sin(bt) + d

is fitted to the data.

Find the value of

b

, correct to 4 decimal places.

Q11·Challenging

The average monthly temperature (°C) in a southern-hemisphere city is modelled by

T(m) = 12\cos\!\left(\dfrac{2\pi(m-1)}{12}\right) + 20

, where

m = 1

is January (the warmest month).

How many months of the year have an average temperature strictly above

26

°C?

Explanation

T(m) > 26 \implies 12\cos\!\left(\frac{\pi(m-1)}{6}\right) > 6 \implies \cos\!\left(\frac{\pi(m-1)}{6}\right) > 0.5

\cos\theta > 0.5

when

\theta \in \left(-\dfrac{\pi}{3},\, \dfrac{\pi}{3}\right)

.

Check each month

m = 1

12

$m$	$\theta = \dfrac{\pi(m-1)}{6}$	$\cos\theta$	$T$ (°C)	$T > 26$ ?
1	0	1.000	32	Yes
2	$\pi/6$	0.866	30.4	Yes
3	$\pi/3$	0.500	26	No (equal)
4	$\pi/2$	0.000	20	No
$\ldots$	$\ldots$	$\ldots$	$\ldots$	No
11	$5\pi/3$	0.500	26	No (equal)
12	$11\pi/6$	0.866	30.4	Yes

Months with

T > 26

°C: January, February, December — that is

\mathbf{3}

months.

Q12·Challenging

A trigonometric function of the form

y = a\cos(b(t - c)) + d

models the height (metres) of a buoy above the seabed, with the following properties:
- maximum height:

9.4

m, minimum height:

1.6

m
- period:

10

hours
- first maximum at

t = 2

hours

Find the value of

c

Open Math

Modelling periodic phenomena

Functions · MAV-12-02

Name:

Date:

Q1Straightforward

The height of the tide at a harbour is modelled by

H(t) = 3\sin\!\left(\dfrac{\pi t}{6}\right) + 4

, where

H

is the height in metres and

t

is time in hours after midnight.

What is the maximum height of the tide in metres?

Q2Straightforward

The height of the tide is modelled by

H(t) = 3\sin\!\left(\dfrac{\pi t}{6}\right) + 4

, where

t

is in hours.

What is the period of the tide in hours?

Q3Straightforward

The average monthly temperature (°C) in a city is modelled by

T(m) = 12\cos\!\left(\dfrac{2\pi m}{12}\right) + 18

, where

m

is the month number with

m = 0

representing the warmest month.

What is the minimum temperature in °C?

Q4Straightforward

The depth of water (metres) at a pier is modelled by

D(t) = 2.5\sin\!\left(\dfrac{\pi t}{6} - \dfrac{\pi}{3}\right) + 5

, where

t

is time in hours.

State the amplitude of the depth in metres.

Q5Moderate

A Ferris wheel has a maximum height of

30

m and a minimum height of

2

m above the ground. Riders board at the lowest point, and the wheel takes

4

minutes to complete one full revolution.

A model of the form

h(t) = -a\cos(bt) + d

is used, where

t

is time in minutes after boarding.

Find the value of

a

Q6Moderate

The temperature

T

(°C) inside a greenhouse follows the model

T(t) = 8\sin\!\left(\dfrac{\pi(t-6)}{12}\right) + 22

, where

t

is hours after midnight,

0 \le t \le 24

.

At what time (hours after midnight) does the temperature first reach its daily maximum? Give an integer answer.

Q7Moderate

The height of the tide is modelled by

H(t) = 1.8\cos\!\left(\dfrac{\pi t}{6}\right) + 3.2

, where

H

is in metres and

t

is hours after midnight.

Find the first time after midnight (to 2 decimal places) when the tide height equals

4

Q8Moderate

The number of hours of daylight in a city in the northern hemisphere is modelled by

D(m) = 3\sin\!\left(\dfrac{\pi m}{6} - \dfrac{\pi}{2}\right) + 12

, where

m

is the month number (January

= 1

, December

= 12

).

Find the number of hours of daylight in April (

m = 4

Q9Moderate

Using the daylight model

D(m) = 3\sin\!\left(\dfrac{\pi m}{6} - \dfrac{\pi}{2}\right) + 12

(northern hemisphere, January

= 1

), find the month number

m

in which daylight is longest.

Q10Challenging

Tide measurements at a wharf give the following data:

$t$ (hours after midnight)	$0$	$3$	$6$	$9$	$12$
$H$ (metres)	$5$	$8$	$5$	$2$	$5$

A sine model

H(t) = a\sin(bt) + d

is fitted to the data.

Find the value of

b

, correct to 4 decimal places.

Q11Challenging

The average monthly temperature (°C) in a southern-hemisphere city is modelled by

T(m) = 12\cos\!\left(\dfrac{2\pi(m-1)}{12}\right) + 20

, where

m = 1

is January (the warmest month).

How many months of the year have an average temperature strictly above

26

°C?

Q12Challenging

A trigonometric function of the form

y = a\cos(b(t - c)) + d

models the height (metres) of a buoy above the seabed, with the following properties:
- maximum height:

9.4

m, minimum height:

1.6

m
- period:

10

hours
- first maximum at

t = 2

hours

Find the value of

c

Worked solutions and answers at openmath.au/year-12/advanced/further-graph-transformations-and-modelling/modelling-periodic-phenomena