6.5 6.5 6.5 6.5 Day 2 Day 2 Day 2 Day 2 Applications Applications Applications Applications ---- Sinusoidal Functions as Mathematical ModelsSinusoidal Functions as Mathematical ModelsSinusoidal Functions as Mathematical ModelsSinusoidal Functions as Mathematical Models

Ex. 1:Ex. 1:Ex. 1:Ex. 1: Jane takes her turn on the vine to practice her swing. As she swings, she goes back and forth across

the river bank from two trees (each branch she lands on is at equal height). She tells Tarzan to set the

stopwatch to take measurements. Assume that her distance varies sinusoidally with the time of her

swing. A quarter of the way into her swing, after 2 seconds, she is 50 feet above the ground. Tarzan finds

that she is at her lowest point 12 feet above the land.

1. Sketch a graph of two

cycles of the swing

2. Find the particular equation expressing 3) Where is Jane at 3.5 seconds?

distance in terms of time.

Ex. 2:Ex. 2:Ex. 2:Ex. 2: When a spaceship is fired into orbit from a site that is not on the equator it goes into an orbit that

takes it alternately north and south of the equator. Its distance from the equator is approximately

sinusoidal function of time. Suppose that a spaceship is fired into orbit from Cape Canaveral. Ten

minutes after it leaves, it reaches its farthest distance north of the equator, 4000 kilometers. Half a

cycle later it reaches its farthest distance south of the equator also 4000 kilometers. The spaceship

completes an orbit once every 90 minutes.

2. Find the particular equation expressing 3. Predict the distance of the spaceship

distance in terms of time. from the equator at 163 minutes.

Ex. Ex. Ex. Ex. 3333:::: A weight attached to the end of a long spring is bouncing up and down.

As it bounces, its distance from the floor varies sinusoidally with time. You

start a stopwatch. When the stopwatch reads 0.3 seconds, the weight first

reaches a high point 60 cm above the floor. The next low point, 40 cm above

the floor, occurs at 1.8 seconds.

a. Sketch a graph of this sinusoidal function. Be sure to include numbers!

b. Write the particular equation expressing c. Predict the distance from the floor when

distance from the floor in terms of the number the stopwatch reads 17.2 seconds.

of seconds the stopwatch read.

d. What was the distance from the floor when e. What were the first two times the spring

you started the stopwatch? was 45 cm above the floor?

Ex. 4Ex. 4Ex. 4Ex. 4 The electricity supplied to your house is called

“alternating current” because the current varies

sinusoidally with time. The frequency of the sinusoid

is 50 cycles per second. Suppose that when t = 0

seconds the current is at its maximum, i = 5 amperes.

a. Write the equation of the sinusoid

b. What is the current when t = 0.01? c. What are the first four times that the current

reaches 4 amperes?

0.020.020.020.02 0.030.030.030.03

−5−5−5−5

5555

x

y

Graphing Sinusoids Quiz – Study Guide (graph at least two periods of each)

1. Graph the following: y 2sin x 36 3

π π= − +

Amp:

Sin. Axis

Period:

Phase Shift:

Start:

End:

2. Graph the following: y 3sec(2x 5 ) 5= − − π − Amp:

Sin. Axis

Period:

Phase Shift:

Start:

End:

3. Graph the following: y 2tan(3 120) 2= θ− + Amp:

Sin. Axis

Period:

Phase Shift:

Start:

End:

Write an equation for each graph below.

4) 5)

y = ____________________ y = ______________________

6) As you stop your car at a traffic light, a pebble becomes wedged between the tire treads. When you

start off, the distance of the pebble from the pavement varies sinusoidally with the distance

you have traveled. The period is, of course, the circumference of the wheel. Assume that the diameter of

the wheel is 24 inches.

a) Sketch a graph of this function b) Write the particular equation

representing this function

c) Predict the distance from the pavement d) When is the first time the pebble is 20

when you have gone 15 inches. inches above the pavement?

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