5.6 Use Sinusoidal Functions to Model Periodic Phenomena Not Involving Angles
Functions 11 McGraw-Hill
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Modelling Alternating Current with Sine Function
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Modelling Predator Prey Populations using trig functions
In the Investigate, the sinusoidal function h(t) = 5 sin [30(t + 3)] is used to model the height of tides in a particular location on a particular day. On a different day, the maximum height is 8 m, the minimum height is -8 m, and high tide occurs at 5:30 am.
a) Modify the function such that it matches the new data.
b) Predict the times for the next high and low tides.
Suppose that a cosine function is chosen to model the tides in the Investigate. Modify the function so that a cosine function is used but all predictions of tides remain the same.
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, to the nearest 50. 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.
The depth, d, in metres, of water in a seaplane harbour on a given day can be modelled using the function d = 12\sin[30(t - 5)] + 14, where t is the time past midnight, in hours.
(a) Determine the maximum and minimum depths of the water in the harbour.
(b) What is the period of the function?
(c) Graph the water level over 24h.
(d) If the water is less than 3m deep, lading a seaplane is considered unsafe. During what time intervals, between midnight and midnight the following day, is it considered unsafe to land a seaplane?
(e) What other factors are important in deciding whether it is safe to land?
At another time of the year, the depth, d, in meters the maximum water depth is 22m and the minimum depth is 6 m. The first high tide occurs at 5:00 am.
(a) Modify the d = 12\sin[30(t - 5)] + 14 to match the new data.
(b) Graph the water level over 24h.
(c) During what time intervals, between midnight and midnight the following day, is it considered unsafe to land a seaplane?
The electricity standard used in Europe and many other parts of the world is alternating current (AC) with a frequency of 50 Hz and a maximum voltage of 240V. The voltage an be modelled as a function of time using a sine function.
(a) What is the period of 50 Hz AC?
(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.
The population of prey can be modelled with the function N(t) = 250 \sin 90t + 500. Research shows that the population of predators follows the period of the population of prey, with a phase shift of \frac{1}{4} of a cycle to the right. Suppose that the predators have a minimum population of 50 and a maximum population of 100.
(a) Construct a model for the population of predators using a sine function.
(b) Graph the population of prey and predators on the same set of axes over a period of 12 years from the base year.
(c) Suggest reasons why there is a phase shift to the right.
The propeller of a small airplane has an overall length of 2.0m. The propeller clears the ground by a distance of 40 cm and spins at 1200 revolutions per minute while the airplane is taxiing.
(a) Model the height of one of the propeller tips above the ground as a function of time using a sinusoidal function.
(b) Graph the function over four cycles.
(c) Determine all times in the first cycle when the tip is 1.0m above the ground.
The wind turbine at Exhibition Place in Toronto is 94 m tall and has there blades, each measuring 24 m in length. The blades turn at a frequency of 27 revolutions per minute.
(a) Use a sinusoidal function to model the height above the ground of one of the blade tips as a function of time.
(c) Determine all times in the first cycle when the tip of the blade is 100 m above the ground.
The notation 11! means to multiply all the natural numbers from the number 11 down to 1. For example, 5! = 5\times 4 \times 3 \times 2 \times 1. When 50! is expanded, how many zeros are at the end of the number?