Sinusoidal Curves Group Activity

Note to the teacher: Each of these activities is designed to be completed by a collaborative group of students. You may prefer to assign one of these problems at a time rather than all of them at once.

Steamboat Problem:

Mark Twain sat on the deck of a river steamboat. As the paddlewheel turned, a point on the paddle blade moved in such a way that its distance, d, from the water's surface was a sinusoidal function of time. When his stopwatch read 4 s, the point was at its highest, 16 ft above the water's surface. The wheel's diameter was 18 ft, and it completed a revolution every 10 s.

1. Sketch a graph of the sinusoidal curve.
2. Find the amplitude, phase shift, period changes, and translations.
3. Write the equation of the graph.
4. Predict the height or depth of the point from the surface of the water when Mark's stopwatch read:
a) 5 s
b) 17 s
c) 0 s
d) 4 s

5. Write at least a half-page explaining how you arrived at the equation and the various heights in question 4.

Fox Population Problem:

Naturalists find that the populations of some kinds of predatory animals vary periodically. Assume that the population of foxes in a certain forest varies sinusoidally with time. Records started being kept when time 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.

1. Sketch a graph of this sinusoidal curve.
2. Find the amplitude, phase shift, period changes, and translations.
3. Write an equation of the graph expressing the number of foxes as a function of time.
4. Predict the population at the following times:
a) t = 7 years
b) t = 0 years
c) t = 9.5 years
d) t = 2 years.

5. Write at least a half-page explaining how you arrived at the equation and the various years in # 4.

Extraterrestrial Being Problem:

Researchers find a creature from an alien planet. Its body temperature is varying sinusoidally with time. 35 minutes after they start timing, it reaches of 120 degrees Fahrenheit. 20 minutes after that it reches its next low, 104 degrees Fahrenheit.

1. Sketch a graph of this sinusoidal curve.
2. Find the amplitude, phase shift, period changes, and translations.
3. Write an equation expressing temperature in terms of minutes since they started timing.
4. Predict the temperature at each of the following times:
a) 40 minutes
b) 0 minutes
c) 75 minutes
d) 10 minutes

5. Write at least a half-page explaining how you arrived at the equation and the various temperatures in # 4.

Submitted by Debbie Lancaster from Lakeside High School in Evans, GA.