More Periodic Functions
 

1. Steamboat Problem:  Mark Twain sat on the deck of a river steamboat.  As the paddle wheel turned, Mr. Clemens noticed a piece of vegetation was caught on a particular point on one of the paddles.  He started to keep track of the time and position above the water of the green object.  When his stopwatch read 4 seconds, the weed was at its highest point, 16 ft above the surface of the water.  The wheels diameter was 18 ft and it completed its revolution every 10 seconds.
   a) Sketch the graph of the height of the weed over time. 
   b) Find the equation of the curve.
   c) How far above the surface was the weed after.  i) 5 seconds  ii) 17 seconds
   d) When was the first time the piece actually touched the surface of the water?
   
   
2. Fox population:  Naturalist find that the populations of some animals varies periodically with time.  Assume the population follows the pattern of a sine curve.  Records started being taken at t = 0 years.  A minimum number, 200 foxes, occurred when t = 2.9 years.  The next maximum, 800 foxes, occurred at t = 5.1 years.
   a) Sketch the graph of this sinusoid.
   b) Write an equation expressing the number of foxes as a function of time.
   c) Foxes are considered endangered when the population drops below 300.  What two nonnegative values of t were foxes first endangered?
   
   
3. Bouncing Spring Problem:  A weight attached to a long spring is bouncing up and down.  As it bounces, its distance from the floor varies periodically with time.  You start a stopwatch.  When the stopwatch reads 0.3 seconds, the weight reaches its first high point 60 cm above the ground.  The next low point, 40 cm above the ground, occurs at 1.8 seconds.
   a) Sketch a graph of the function.
   b) Write an equation expressing the distance above the ground in terms of the numbers of seconds the stopwatch reads.
   c) What is the distance above the ground after 17.2 seconds?
   
   
4. Sunspot Problem:  For several hundred years, astronomers have kept track of the number of solar flares, or "sunspots," which occur on the surface of the sun.  The number of sunspots counted in a given year varies from a minimum of about 10 per year to a maximum of 110 per year.  Between the maximums that occurs in the years 1750 and 1948, there were 18 complete cycles.
   a) What is the period of the sunspot cycle?
   b) Assume that the number of sunspots varies sinusoidally with the year.  Sketch a graph of two sunspot cycles, starting in 1948.
   c) Write an equation expressing the number of sunspots per year in terms of the year.
   d) How many sunspots would you expect this year?
   
   
5. Tide Problem: Suppose that you are on Drakes Beach in Point Reyes.  At 2:00 P.M. on October 2, the tide is (the water is at its deepest).  At that time you find that the water at the end of a pier is 1.5 meters.  At 8:00 P.M. the same day, when the tide is out, the water is at 1.1 meters.  Assume that the depth of the water varies sinusoidally with time.
   a)  Sketch a graph of the tide from 12 noon on October 2
   b) Find an equation of the function.
   c) Use your equation to predict the depth of the water at: 
          i) 4 P.M. (Oct 2) ii) 5 P.M.on Oct 3
   
   
6. Tidal Wave Problem: A tsunami (a tidal wave) is a fast moving ocean wave caused by an underwater earthquake.  The water first goes down from its normal level and than 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 meters approaches the pier at Honolulu, where the normal depth of the water is 9 meters. Assuming that the depth of the water varies sinusoidally with time as the tsunami passes.
   a)Sketch the graph of the function of the height of that water at the pier.  The cycle starts at its normal depth (9 meters).
   b) Find an equation for the function.
   c) What is the water height at i) 2 min, ii) 4 min and iii) 12 min.
   d) Between what two times is their no water at the pier.
   
   
7. Biorhythm Problem:  According to biorhythm theory, your body is governed by three independent sinusoidal functions, each with a different period as follows:
        Physical function:  Period = 23 days
        Emotional function:  Period = 28 days
        Intellectual function:  Period = 33 days
   a)  Phoebe Small is at a high point on all three cycles today!  This means that she is at the very highest ability in all three areas. Assume the amplitude is 100 units and each graph has no vertical shift. Sketch graphs of all three cycles on the same axes.
   b) Write equations for each of the three functions. 
   c) Phoebe will be at her intellectual peak in 33 days.  What are the values of the other two cycles on this day?
   
   
8. Shock-Felt-Round-the-World:  Suppose that one day all 200 million people in the United States climb up on tables.  At time = 0, we all jump off.  The resulting shock as we hit the earth's surface will start the entire earth vibrating in such a way that the earth first moves down from its normal position and then moves up an equal distance above its normal position.  The displacement is a sinusoidal function of time with a period of 54 minutes.  Assume that the amplitude is 50 meters.  
   a) Sketch a graph of the function.
   b) Find an expression of the displacement in terms of the time elapsed.
   c) What is the displacement after 21 seconds?
   d) When will be the first two times