11B Topic 4_2

6. Applications of periodic functions

Challenge Question (1)

High tide is 4.5 m at midnightLow tide is 0.5m at 6am

i) Find the height of the tide at 7pm?ii) Between what times will the tide be greater

than or equal to 3m?

Use y = A cos B(x+C) + D

i) Find “A”

Tide range = 4.5 - 0.5 = 4 A = 2 y = 2cos B(x+C) + D

iii) Find “B”

Period = 12

ii) Find “D”

D = 4.5 – 2 = 2.5

y = 2cos B(x+C) + 2.5

2Period=

212

6

2cos 2.56

B

B

B

y x C

iv) Find “C”

We can see from the graph that no C-value is needed

High tide is 4.5 m at midnight Low tide is 0.5m at 6ami) Find the height of the tide at 7pm?ii) Between what times will the tide be greater than or equal

to 3m?

2cos 2.56xy

2cos 2.56xy

By use of TI calculator…

i) What is the tide height at 7pm?

• Graph using suitable windows• 2nd Calc option 1. Value• Enter 19• Answer = 0.77m (2D.P.)

ii) Tide above 3m• Add y = 3 to the graph• 2nd Calc option 5.

Intersect• Follow prompts• Answer = • MN – 2:31am • 9:29am – 2:31pm• 9:29pm – MN

Challenge Question (2)

High tide of 4.2m occurs in a harbor at 4am Tuesday and the following low tide of 0.8m occurs 6¼ hours later. If a ship entering the harbor needs a minimum depth of water of 3m, what times on Tuesday can this vessel enter?

Model: The graph below shows the horizontal displacement of a pendulum from its rest position over time:

(a) Find the period and amplitude of the movement.(b) Predict the displacement at 10 seconds.(c) Find all the times up to 20 sec when the displacement will be 5 cm to the

right (shown as positive on the graph)

X

Y

1 2 3 4 5

-8-6-4-2

2468

0

Model: The graph below shows the horizontal displacement of a pendulum from its rest position over time:

(a) Find the period and amplitude of the movement.(b) Predict the displacement at 10 seconds.(c) Find all the times up to 20 sec when the displacement will be 5 cm to the

right (shown as positive on the graph)

X

Y

1 2 3 4 5

-8-6-4-2

2468

0

Period = 4.5 - 0.5

= 4 sec

Model: The graph below shows the horizontal displacement of a pendulum from its rest position over time:

(a) Find the period and amplitude of the movement.(b) Predict the displacement at 10 seconds.(c) Find all the times up to 20 sec when the displacement will be 5 cm to the

right (shown as positive on the graph)

X

Y

1 2 3 4 5

-8-6-4-2

2468

0

Amplitude = 8

Model: The graph below shows the horizontal displacement of a pendulum from its rest position over time:

(a) Find the period and amplitude of the movement.(b) Predict the displacement at 10 seconds.(c) Find all the times up to 20 sec when the displacement will be 5 cm to the

right (shown as positive on the graph)

X

Y

1 2 3 4 5

-8-6-4-2

2468

0

Since the period = 4 sec

Displacement after 10 sec will be the same as displacement after 2 sec

= 5.7cm to the left

Model: The graph below shows the horizontal displacement of a pendulum from its rest position over time:

(a) Find the period and amplitude of the movement.(b) Predict the displacement at 10 seconds.(c) Find all the times up to 20 sec when the displacement will be 5 cm to the

right (shown as positive on the graph)

X

Y

1 2 3 4 5

-8-6-4-2

2468

0

Displacement= 5cm

t = 1.1

3.9 7.9, 11.9, 15.9, 19.9

5.1, 9.1, 13.1, 17.1

Exercise

NewQ P 179 Set 5.2 1,3

Model: Find the equation of the curve below.

X

Y

1 2 3 4 5 6 7 8 9 10

-2

2

0

Amplitude = 2.5 y = a sin b(x+c)

Model: Find the equation of the curve below.

X

Y

1 2 3 4 5 6 7 8 9 10

-2

2

0

Amplitude = 2.5 y = 2.5 sin b(x+c)

Period = 6

Period = 2/b 6 = 2/b

b = /3

Model: Find the equation of the curve below.

X

Y

1 2 3 4 5 6 7 8 9 10

-2

2

0

Amplitude = 2.5 y = 2.5 sin /3(x+c)

Period = 6

Period = 2/b 6 = 2/b

b = /3

Phase shift = 4 ()

so c = -4

Model: Find the equation of the curve below.

X

Y

1 2 3 4 5 6 7 8 9 10

-2

2

0

Amplitude = 2.5 y = 2.5 sin /3(x-4)

Period = 6

Period = 2/b 6 = 2/b

b = /3

Phase shift = 4 ()

so c = -4

Exercise

NewQ P 183 Set 5.3 1,4

Exercise 5.3 pg 183, No.4

Find the equation of the curve below in terms of the sin function and the cosine

function.