Math 11AD
Sinusoidal Functions
1. Tarzan is swinging back and forth on his grapevine. As he swings, he goes back and forth across the river bank, going alternately over land and water. Jane decides to mathematically model his motion and starts her stopwatch. Let x be the number of seconds the stopwatch reads and let y be the number of meters Tarzan is from the river bank. Assume that y varies sinusoidally with t, and the y is position when Tarzan is over water and negative when he is over land. Jane finds that when t = 2, Tarzan is at one end of the swing, where y = -23. She finds that when t = 5 he reaches the other end of his swing and y = 17.
a. Sketch a graph of this sinusoidal function.
b. Write the particular equation expressing Tarzan’s distance from the river bank in terms of t.
c. Predict y when
i. t = 2.8
ii. t = 6.3
iii. t = 15
2. As you ride the Ferris wheel, your distance from the ground varies sinusoidally with time. Once the ride is filled, you find that it takes you 3 seconds to reach to top, 43 feet above the ground, and the wheel makes a revolution once every 8 seconds. The diameter of the wheel is 40 feet.
a. Sketch a graph of this sinusoid function.
b. What is the lowest you go as the Ferris wheel turns, and why is this number greater than zero?
c. Write the particular equation of this sinusoid.
d. Predict your height above the ground when
i. t = 6
ii. t = 4.3333
iii. t = 0
3. A mass 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 second, the mass 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.
b. Write the equation.
c. Predict the distance from the floor when the stopwatch reads 17.2 seconds.
4. A tsunami is a fast-moving ocean wave caused by an underwater earthquake. The water first goes down from its normal level, then rises an equal distance above its normal level. The period is about 15 minutes. Suppose that a tsunami with an amplitude of 10 meters approaches the pier at Halifax, where the normal depth of the water is 9 meters.
a. Assume that the depth of the water varies sinusoidally with time. Predict the depth of the water when time equals
i. 2 minutes
ii. 4 minutes
iii. 12 minutes
b. According to your model, what will the minimum depth of the water be? How do you interpret this answer in terms of what will happen in the real world?
c. The “wavelength” of a wave is the distance a crest to the wave travels in one period. It is also equal to the distance between two adjacent crests. If a tsunami travels at 1200 kilometres per hour, what is its wavelength?
5. The electricity supplied to your house is called “alternating current: because the current varies sinusoidally with time. The frequency of the sinusoid is 60 cycles per second. Suppose that at time t = 0 seconds the current is at its maximum, i = 5 amperes.
a. Write an equation expressing current in terms of time.
b. What is the current when t = 0.01?
- The wheel radius on a paddlewheel steamer is 4 m.The boat gets maximum performance when the axle is 1 meter above the waterline.A frog swimming under water gets caught on the paddlewheel and starts rotating above the wheel.The frog is first noticed at the top of the wheel.If the frog was at the highest of the wheel every 10 seconds, find an equation:
a) How fast is the frog rotating?
b) What does the horizontal axis of the sketch represent?
7. Question Details:
A waterwheel rotates at 6 revolutions per minute (rpm). After 2 seconds, point A is at a maximum height above the water. How high is the point A when the wheel starts spinning (according to your equation)?