Alexis is swinging back and forth on a trapeze. Her distance from a vertical support beam in terms of time can be modelled by a sinusoidal function.
At 1 second, she is the maximum distance from the beam, 12 m. At 3 seconds, she is the minimum distance from the beam, 4 m.
a) Determine an equation of a sinusoidal function that describes Alexis’s distance from the vertical beam in relation to time.
b) Give the domain and range of the function and explain what they mean in this scenario.
Question 1 Solution
One equation that models this scenario is:
Skyscrapers can sway in high-wind conditions. In one case, at t = 2 seconds, the top floor of a building swayed 30 cm to the left (-30 cm), and at t = 12 seconds, the top floor swayed 30 cm to the right (+30 cm) of its starting position.
a) What is an equation of a sinusoidal function that describes the motion of the building in terms of time?
b) Dampers, in the form 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?
Question 2 Solution
Shreeya is floating in an inner tube in a wave pool. She is 1.5 m from the bottom of the pool when she is at the trough of a wave. A stopwatch starts timing at this point. In 1.25 seconds, she is on the crest of the wave, 2.1 m from the bottom of the pool.
a) Determine the equation of the function that expresses Shreeya’s distance from the bottom of the pool in terms of time.
b) What is the amplitude of this function and what does it represent in this situation?
c) How far above the bottom of the pool is Shreeya at t = 4 seconds?
d) How many complete cycles are there within first 40 seconds?
Question 3 Solutions
b) The amplitude ‘a’ is 0.3. This represents the height of the crest (the wave) relative to it’s normal water level.
c) If t = 4, substitute t = 4 into the equation to solve for ‘d’:
e) 16 cycles
In one area of the Bay of Fundy, the tides cause the water level to rise to 6 m above average sea level and to drop to 6 m below average sea level. One cycle is completed approximately every 12 hours. Assume the changes in the depth of water over time can be modelled by a sinusoidal function.
a) Write an equation for the graph to show how the depth of water changes over the next 24 hours. Assume that at low tide, the depth of the water is 2 m.
b) If the water is at an average sea level at 2:00am, and the tide is coming in, write an equation for the graph that shows how the depth changes over the next 24 hours.
Question 4 Solutions
A Ferris wheel with a radius of 7 m makes one complete revolution every 16 seconds. The bottom of the wheel is 1.5 m above the ground.
a) Find an equation for the relationship between the height and...