5.6 Use Sinusoidal Functions to Model Periodic Phenomena Not Involving Angles
Chapter 5
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Lectures 2 Videos
Solutions 13 Videos

The population of prey in a predator-prey relation is shown. Time is in years since 1985.

a) Determine the maximum and minimum values of the population. Use these to find the amplitude.

b) Determine the vertical shift, c.

c) Determine the phase shift, d.

d) Determine the period. Use the period to determine the value of k.

e) Model the population versus time with a sinusoidal function.

f) Graph your function. Compare it to the graph shown.

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Refer to question 3. Suppose the period was 8 years. How would your equation in part e) change? Write the new equation.

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The depth of the water at the end of a pier is 2 m at low tide and 12 m at high tide. There are 6 h between low tide and high tide. The first high tide occurs 6 h after midnight. One complete cycle takes 12 h.

a) Write a sinusoidal equation that represents the depth of the water.

B) Determine the depth of the water at 4:15 a.m. and 3:30 p.m.

c) Graph your equation from part a) and plot the points represented by you answers in part b).

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The number of sunlight hours on each day of the year for a particular location can be modelled by a periodic function, London, Ontario, the maximum number of sunlight hours, 15.3 h, occurs on June 21. The minimum number of sunlight hours, 9.3 h, occurs on December 21.

a) Use a cosine function to model the number of sunlight hours in London, Ontario. Note that June 21 is day 172 and December 21 is day 355.

b) Determine the number of sunlight hours on April 1.

c) Determine a day when there are 14 h of sunlight.

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The sinusoidal function h(t) = 7 sin [30(t - 2.5)] models the height, h, of tides in a particular location on a particular day at t hours after midnight.

a) Determine the maximum height and the minimum height of the tides.

b) At what times do high tide and low tide occur?

c) Use a cosine function to write an equivalent equation.

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Refer to question 1. On a different day, the maximum height is 4.5 m, the minimum is -4.5 m, and low tide occurs at 5:00 a.m.

a) Modify the sine function so that it matches the new data.

b) Predict the times for the next low and high tides.

c) Modify your equation from part a) so that low tide occurs at 5:30 a.m.

d) Write an equivalent equation using a cosine function for your answer to part c).

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Refer to the Example at the beginning of this section. Use the equation you found in part d) to determine the number of hours of daylight on

a) April 1

b) September 1

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The depth of water, d, in metres, of a seaside inlet on a given day can be modelled by the function d = 7 sin [30(t - 4)] + 11, where t is the time past midnight, in hours.

a) Determine the maximum and minimum depths of water in the inlet.

b) What is the period between maximum values?

c) Graph the water level over a period of 24 h.

d) When is the water 5 m deep?

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Angelina constructs a model alternating current (AC) generator in physics class and cranks it by hand at 5 revolutions per second. She is able to light up a flashlight bulb that is rated for 8 V.

a) What is the period of the AC produced by the generator?

b) Determine the value of k.

c) What is the amplitude of the voltage function?

d) Model the voltage with a suitably transformed sine function.

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The graph shows how the voltage, V, in volts, varies with time, t, in seconds, for the electricity provided to an electrical appliance.

a) Identify the period and amplitude of the function.

b) What is the maximum voltage?

c) Write an equation to model this situation.

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The table shows the temperature readings every 2 h over a 24-h period on an early summer day.

a) Use the data to write an equation that models the temperature over the 24-h period.

b) Explain how you can check the accuracy of your equation.

c) Use your equation to predict the temperature at the following times:

i) 04:00

ii) 16:00

iii) 20:30

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The blades on a windmill turn at a frequency of 20 revolutions per minute. The length of each blade is 3.5 m, and the tips of the blades are 8 m off the ground when they are at their lowest point.

a) Use a sinusoidal function to model the height above the ground of one of the blade tips as a function of time if the blade's height is measured starting from the center of rotation.

b) Graph the function.

c) Determine the times in the first cycle when the tip of a blade is 10 m off the ground.

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Pistons in a vehicle engine move up and down many times each second. The graph shows how the height varies with time for the pistons in a 6-cylinder engine.

a) Determine an equation for each graph.

b) Determine the frequency of the motion.

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