Math 30-2 Formative Practice Test 2 Sinusoidal Functions

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Math 30-2

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Mathematics 30-2 Formula Sheet

Relations and Functions

Graphing Calculator Window Format

Exponents and Logarithms

Laws of Logarithms

Exponential functions

Sinusoidal functions

Quadratic equations

For ,

Probability

, where and .

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Written Response: **SHOW ALL OF YOUR WORK!!!**

1. Consider the sinusoidal function .

a) Determine the amplitude, period, midline, minimum, maximum, domain and range for the function.

Amplitude Period Midline Minimum Maximum Domain Range

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2. During a physics experiment, Sally held down a weight attached to a spring, then released the weight. The height of the weight above the table is shown in the following table.

Time (s) 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Height (in.) 5 11 17 11 5 11 17 11 5

a) Determine the sinusoidal regression equation in the form . (Round to 2 decimal places)

b) Use your equation from part a) to determine the height of the after 0.75 seconds. (If you were unable to answer part a), use the equation )

c) Use your equation from part a) to determine the time at which the weight will first reach a height of 13 inches. (If you were unable to answer part a), use the

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1. Sketch the two functions in the space below.

 Day 345 is the shortest day of the year. Label this day on the graph with an X.

 Determine the number of hours of daylight, to the nearest hundredth, on the shortest day of the year.

 George leaves for work at 7:30 each morning. Explain how to find the number of days in one year (365 days) that George will leave the house when it is dark.

X: [ 0, 730, 100 ] Y: [ , , ]

The times of sunrise and sunset in Medicine Hat, Alberta, during the year 2008 can be found using the following sinusoidal equations, where x is the number of the day of the year (1 is January 1, 2 is January 2, 3 is January 3, and so on) and y is the time of the sunrise or sunset, expressed as a decimal fraction of an hour.

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2. State the maximum and minimum heights of the wheel and the period of one rotation. Maximum _______ Minimum _______ Period _______

 Graph the height of the end of the paddle wheel as it goes through 4 rotations. Start the graph where the height is 6 m.

 What is the shallowest river the paddle wheel can go in without scraping the bottom?

 Write a sinusoidal function, in the form

y

=

a

sin

(

bx

+

c

)

+

d

, to represent the height of the end of the paddle wheel, y, for time, x. Use c = 1.57.

 The paddle wheel is replaced with a new wheel. The new wheel has a radius of 5 m and is attached 3 m above the surface of the water. The new wheel still completes 6 rotations in 60 seconds. Explain how the equation would change to reflect this information.

Time (s) Height

(m)

0 5 10 15 20 25 30 35 40

10 8 6 4 2 0 -2 -4

water surface = 0.0 m

r = 6.0 m

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Numerical Response (4 marks)

Numerical Response 1:

To the nearest degree, 6.04 radians is equivalent to __________. (Place your answer in the boxes from left to right)

Use the following information to answer the next question.

The height of a pendulum, h, in inches, above a table top t seconds after the pendulum is released can be modeled by the sinusoidal regression function

To the nearest tenth of an inch, the height of the pendulum at the moment of release is _____________.

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Consider the sinusoidal function

The period, to the nearest hundredth, of the function is ____________. (Place your answer in the boxes from left to right)

The minimum, to the nearest tenth, of the function is ____________.

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Multiple Choice (8 marks)

1. When converted to radians, 80° is equivalent to: A.

B.

C.

D.

The average daily high temperature of Montreal, in °F, for each of the months of the year is shown in the table below. January is month 1, February is month 2, etc..

Month Average Daily High

Temperature (°F) Month Average Daily HighTemperature (°F)

1 22 7 80

2 25 8 77

3 36 9 67

4 52 10 51

5 66 11 41

6 75 12 28

This information can be modeled using a sinusoidal regression function in the form .

2. The value of d, to the nearest hundredth, is: A. -2.02

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Use the following information to answer the next questions.

The height of a rider on a Ferris wheel can be modeled by the sinusoidal regression function

where h is the height of the rider above the ground, in metres, and t is the time in minutes after the ride starts.

3. According to the sinusoidal regression function, the maximum height of the rider above the ground is:

A. 14 m B. 8 m C. 6 m D. 2 m

4. The time it takes for the rider to reach a height of 10 metres for the first time is: A. 1.0 min

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Use the following information to answer the next question. The graph of a sinusoidal function is shown below.

5. The amplitude of the sinusoidal function is ____i____ units and the midline is y = ____ii____ units.

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The partial graph of a sinusoidal function of the form where is shown below.

6. The value of b is: A. 8

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The height of a student above the ground on a circular Ferris Wheel can be modeled by the equation , where H(t) is the height in metres after t seconds.

The diameter of the wheel is 22 metres and the student starts at the bottom of the Ferris Wheel at t = 0 at a height of 4 metres above the ground.

The Ferris Wheel reaches a maximum height for the first time after 45 seconds.

7. The value of d in the equation is: A. 22

B. 15 C. 14 D. 11

8. The value of b in the equation is: A. 90

B. 45 C.

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Replacement Question (Optional)

This question is worth one mark towards your total and may be completed in place of another question. If you choose to replace a question please note that the original question will not be counted.

If you are using this as a replacement question, please tell me what question you are choosing to replace, and why you are choosing to replace it.

 Replacement Question _______________________________________(i.e. WR 1, bullet 3 or MC 5)  Why did you choose to replace this question?

 The first Ferris Wheel every built was created by a bridge builder by the name of George W. Ferris in 1893.

The diameter of the wheel was approximately 76 metres and the maximum height of of the Ferris Wheel was approximately 80 metres.

The wheel rotated one full revolution every nine minutes.

Determine a sinusoidal equation of the form to represent this situation.

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Solutions:

Written Response Solutions: See website

Numerical Response: 1. 346

2. 3.3 3. 3.49 4. 5.1

Multiple Choice 1. C