6.7 Solving Problems Using Sinusoidal
Chapter
Chapter 6
Section
6.7
Solutions 26 Videos

The load on a trailer has shifted, causing the rear end of the trailer to swing left and right. The distance from one of the tail lights on the trailer to the curb varies sinusoidally with time. The graph models this behaviour.

a) What is the equation of the axis of the function, and what does it represent in this situation?

b) What is the amplitude of the function, and what does it represent in this situation?

c) What is the period of the function, and what does it represent in this situation?

d) Determine the equation and the range of the sinusoidal function.

e) What are the domain and range of the function in terms of the situation?

f) How far is the tail light from the curve at t 5 3.2 s?

2.58mins
Q1

Dan Ellis, a fictional character in a French novel, attacked windmills because he thought they were giants. At one point, he got snagged by one of the blades and was hoisted into the air. The graph shows his height above ground in terms of time.

• What is the equation of the axis of the function, and what does it represent in this situation?
0.52mins
Q2a

Dan Ellis, a fictional character in a French novel, attacked windmills because he thought they were giants. At one point, he got snagged by one of the blades and was hoisted into the air. The graph shows his height above ground in terms of time.

• What is the amplitude of the function, and what does it represent in this situation?
0.30mins
Q2b

Dan Ellis, a fictional character in a French novel, attacked windmills because he thought they were giants. At one point, he got snagged by one of the blades and was hoisted into the air. The graph shows his height above ground in terms of time.

• What is the period of the function, and what does it represent in this situation?
0.17mins
Q2c

Dan Ellis, a fictional character in a French novel, attacked windmills because he thought they were giants. At one point, he got snagged by one of the blades and was hoisted into the air. The graph shows his height above ground in terms of time.

• If Don Quixote remains snagged for seven complete cycles, determine the domain and range of the function.
0.59mins
Q2d

Dan Ellis, a fictional character in a French novel, attacked windmills because he thought they were giants. At one point, he got snagged by one of the blades and was hoisted into the air. The graph shows his height above ground in terms of time.

• Determine the equation of the sinusoidal function.
0.45mins
Q2e

Dan Ellis, a fictional character in a French novel, attacked windmills because he thought they were giants. At one point, he got snagged by one of the blades and was hoisted into the air. The graph shows his height above ground in terms of time.

• If the wind speed decreased, how would that affect the graph of the sinusoidal function?
0.24mins
Q2f

Chris is swinging back and forth on a trapeze. His distance from a vertical support beam in terms of time can be modelled by a sinusoidal function. At 1 s, he is the maximum distance from the beam, 12 m. At 3 s, he is the minimum distance from the beam, 4 m. Determine an equation of a sinusoidal function that describes Chris' distance from the vertical beam in relation to time.

3.08mins
Q3

The interior and exterior temperatures of an igloo were recorded over a 48 hour period. The data were collected and plotted, and two curves were drawn through the appropriate points.

• How are these curves similar? Explain how each of them might be related to this situation.
1.07mins
Q4a

The interior and exterior temperatures of an igloo were recorded over a 48 h K period. The data were collected and plotted, and two curves were drawn through the appropriate points.

• i. Describe the domain and range of each curve.
• ii. Assuming that the curves can be represented by sinusoidal functions, determine the equation of each function.
2.23mins
Q4bc

Skyscrapers sway in high-wind conditions. In one case, at t= 2s, the top floor of a building swayed 30 cm to the left (-30 cm), and at t= 12, the top floor swayed 30 cm to the right (+30 cm) of its starting position.

• What is the equation of a sinusoidal function that describes the motion of the building in terms of time?
2.04mins
Q5a

Skyscrapers sway in high-wind conditions. In one case, at t= 2s, the top floor of a building swayed 30 cm to the left (-30 cm), and at t= 12s, the top floor swayed 30 cm to the right (+30 cm) of its starting position.

• Dampers, in the forms of large tanks of water, are often added to the top floors of skyscrapers to reduce the severity of the sways. If a damper is added to this building, it will reduce the sway (not the period) by 70\%. What is the equation of the new function that describes the motion of the building in terms of time?
0.31mins
Q5b

Mel is floating in an inner tube in a wave pool. He is 1.5 m from the bottom of the pool when he is at the trough of a wave. A stopwatch starts timing at this point. In 1.25 s, he is on the crest of the wave, 2.1 m from the bottom of the pool.

• Determine the equation of the function that expresses Milton’s distance from the bottom of the pool in terms of time.
1.58mins
Q6a

Mel is floating in an inner tube in a wave pool. He is 1.5 m from the bottom of the pool when he is at the trough of a wave. A stopwatch starts timing at this point. In 1.25 s, he is on the crest of the wave, 2.1 m from the bottom of the pool.

• What is the amplitude of the function, and what does it represent in this situation?
0.34mins
Q6b

Mel is floating in an inner tube in a wave pool. He is 1.5 m from the bottom of the pool when he is at the trough of a wave. A stopwatch starts timing at this point. In 1.25 s, he is on the crest of the wave, 2.1 m from the bottom of the pool.

• How far above the bottom of the pool is Milton at t = 4 s?
0.24mins
Q6c

Mel is floating in an inner tube in a wave pool. He is 1.5 m from the bottom of the pool when he is at the trough of a wave. A stopwatch starts timing at this point. In 1.25 s, he is on the crest of the wave, 2.1 m from the bottom of the pool.

*f data are collected for only 40 s, how many complete cycles of the sinusoidal function will there be?

0.29mins
Q6d

Mel is floating in an inner tube in a wave pool. He is 1.5 m from the bottom of the pool when he is at the trough of a wave. A stopwatch starts timing at this point. In 1.25 s, he is on the crest of the wave, 2.1 m from the bottom of the pool.

• If the period of the function changes to 3 s, what is the equation of this new function?
0.44mins
Q6e

An oscilloscope hooked up to an alternating current (AC) circuit shows a sine curve. The device records the current in amperes (A) on the vertical axis and the time in seconds on the horizontal axis. At t = 0 s, the current reads its first maximum value of 4.5 A. At t =\frac{1}{120} s, the current reads its first minimum value of -4.5 A. Determine the equation of the function that expresses the current in terms of time.

2.39mins
Q7

Christine is holding on to the end of a spring that is attached to a lead ball. As she moves her hand slightly up and down, the ball moves up and down. With a little concentration, she can repeatedly get the ball to reach a maximum height of 20 cm and a minimum height of 4 cm from the top of a surface. The first maximum height occurs at 0.2 s, and the first minimum height occurs at 0.6 s.

(a) Determine the equation of the sinusoidal function that represents the height of the lead ball in terms of time.

(b) Determine the domain and range of the function.

(c) What is the equation of the axis, and what does it represent in this situation?

(d) What is the height of the lead ball at 1.3 s?

4.26mins
Q8

A paintball is shot at a wheel of radius 40 cm. The paintball leaves a circular mark 10 cm from the outer edge of the wheel. As the wheel rolls, the mark moves in a circular motion.

• Assuming that the paintball mark starts at its lowest point, determine the equation of the sinusoidal function that describes the height of the mark in terms of the distance the wheel travels.
2.53mins
Q9a

A paintball is shot at a wheel of radius 40 cm. The paintball leaves a circular mark 10 cm from the outer edge of the wheel. As the wheel rolls, the mark moves in a circular motion.

• If the wheel completes five revolutions before it stops, determine the domain and range of the sinusoidal function.
0.59mins
Q9b

A paintball is shot at a wheel of radius 40 cm. The paintball leaves a circular mark 10 cm from the outer edge of the wheel. As the wheel rolls, the mark moves in a circular motion.

• What is the height of the mark when the wheel has travelled 120 cm from its initial position?
0.43mins
Q9c

The population of rabbits, R(t), and the population of foxes, F(t), in a given region are modelled by the functions R(t) = 10 000 1 5000 \cos(15t)^o and F(t) = 1000 + 500 \sin(15t)^o, where t is the time in months. Referring to each graph, explain how the number of rabbits and the number of foxes are related.

2.39mins
Q10

Two pulleys are connected by a belt. Pulley A has a radius of 3 cm, and Pulley B has a radius of 6 cm. As Pulley A rotates, a drop of paint on the circumference of Pulley B rotates around the axle of Pulley B. Initially, the paint drop is 7 cm above the ground. Determine the equation of a sinusoidal function that describes the height of the drop of paint above the ground in terms of the rotation of Pulley A.

Q12

Examine the graph of the function f(x).

a) Determine the equation of the function.

b) Evaluate f(20).

c) If f(x) = 2, then which of the following is true for x?

• i) 180^o + 360^ok, k\in \mathbb{I}
• ii) 360^o + 180^ok, k\in \mathbb{I}
• iii) 90^o + 180^ok, k\in \mathbb{I}
• iv) 270^o + 360^ok, k\in \mathbb{I}

d) If f(x) = -1, then which of the following is true for x?

• i) 180^o + 360^ok, k\in \mathbb{I}
• ii) 360^o + 900^ok, k\in \mathbb{I}
• iii) 90^o + 360^ok, k\in \mathbb{I}
• iv) 90^o + 180^ok, k\in \mathbb{I}
Without using graphing technology, determine x when f(x) = 7 for the function f(x) =4\cos(2x) +3 in the domain \{x \in \mathbb{R}\vert 0^o\leq x \leq 360^o\}.