Applied 40S Applications of Sinusoidal Functions

1. As you stop your car at a traffic light, a pebble becomes wedged between the tire treads. As you start to drive, the pebble remains stuck in the tire tread, and the distance of the pebble from the pavement varies sinusoidally with the distance you drive. The period is, of course, the circumference of the wheel, and the wheel has a diameter of 24 inches. (The graph starts when the pebble meets the tire).

a. Sketch a graph of this function. b. Write the sinusoidal equation of this function. c. Calculate the distance the pebble is from the pavement after you have

driven 15 inches; 100 inches

2. Cottage owner Brad (on Vancouver Island) measures the depth of the water at his dock 10 times during the course of one day. The water level keeps changing because of the tides. The times are based on the 24-hour clock, and are written in decimal form (i.e., 5:45 AM is 5.75 hours). The chart shows the times and the water depths.

Time(h) 5.75 6.50 7.50 8.25 9.5 11.00 12.50 13.50 15.25 16.00

Depth(m) 2.45 2.77 3.54 3.99 3.83 2.72 2.61 3.32 4.04 3.69

a. Sketch a rough graph.b. Use your calculator to determine an equation to represent the data. Round

the values of the parameters to two decimal places. c. What is the depth of the water at 23:00 hours (i.e., 11 PM)? d. What is the median depth of the water? e. How much does the water depth vary from the lowest water level to the

highest?

3. On May 3rd, the depth of the water in an east coast harbour will vary over time as described by the equation:

y = 2.3 sin 0.506(x + 3.1) + 2.8

where 'x' represents the time (hours), and 'y' represents the depth (metres) of the water. The time at x = 0 is midnight of May 2nd, and x = 12 is noon of May 3rd.

a. What are the minimum and maximum depths of water in the harbour? b. What is the average depth of the water in the harbour? c. How much time is there between two high tides? d. What is the depth of the water at 8:00 AM? e. At what time in the afternoon is the water at it's lowest? How deep is the

water at this time? (This is the time that little Ziggi likes to go kayaking in the harbour because there are no large boats moving at this time.)

f. A commercial boat requires at least three metres of water to move around the harbour. Describe how you would determine when it is safe for the boat to operate in the harbour.

4. A tsunami (commonly called a "tidal wave" because its effect is like a rapid change in tide) 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, and finally returns to its normal level. The period is about 15 minutes. Suppose that a tsunami with an amplitude of 10 metres approaches the pier at Honolulu, where the normal depth of the water is 9 metres.

a. Assuming that the depth of the water varies sinusoidally with time as the tsunami passes, predict the depth of the water at the following times after the tsunami first reaches the pier.

i. 2 minutes ii. 4 minutes

iii. 12 minutes b. According to your model, what will be the minimum depth of the water?

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 of the wave travels in one period. It is also equal to the distance between two adjacent crests. If a tsunami travels at 800 kilometres per hour, what is its wavelength?

5. The chart below shows the number of hours of daylight in Winnipeg for certain days of a year.

Day of the Year 7 38 67 98 128 159

No. of Hours 8.317 9.667 11.417 13.333 15.067 16.217

 

Day of the Year 189 220 251 281 312 342

No. of Hours 16.133 14.85 13.067 11.217 9.417 8.233

a. Using sine regression, write a sinusoidal equation that represents the data. Round the values of the parameters to three decimal places.

b. What is the average length of a day (i.e., sunlight hours) in Winnipeg? Round your answer to the nearest minute.

c. Which day of the year is the longest day? d. What is the length of the longest day of the year? Write your answer

rounded to the nearest minute.