Sinusoidal Functions as Mathematical Models
For each situation,
 Sketch at least two periods of the graph.
 Label maximum, minimum, and “mid-line” points on your sketch.
 Determine the amplitude, period, phase shift, and vertical shift.
 Write an equation to model the situation.
 Determine any requested function values.
1. A weight 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 seconds, the weight first reaches
a high point 60 cm above the floor. The next low point, 40 cm above the floor, occurs at 1.8 seconds. What was the
distance from the floor when you started the stopwatch (𝑡 = 0)?
2. Suppose that a water wheel with a 7 foot radius rotates at 6 revolutions per minute. The height of a point P on the
rim of the wheel varies sinusoidally with time in seconds. The water cuts off a 1 foot deep segment at the bottom of
the wheel. You start your stopwatch. Two seconds later, point P is at its greatest height. Predict the distance above
the water line that point P will be after 5.5 seconds.
P
water line
1’1'
3. As you stop your car at a traffic light, a pebble becomes wedged between the tire treads. When you start off, the
distance of the pebble from the pavement varies sinusoidally with the distance you have traveled. The period is the
circumference of the wheel. The diameter of the wheel is 24 inches. Predict the distance the pebble is from the
pavement when you have gone 15 inches.
4. Naturalists find that the population of some kinds of predatory animals varies 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 of 200 foxes occurred when 𝑡 = 2.9 years. The next maximum, 800 foxes, occurred at
𝑡 = 5.1 years. Foxes are declared to be an endangered species when their population drops below 300. Between
what two values of 𝑡 were foxes first endangered?
5. You are on board a submarine submerged in the Pacific Ocean. At time 𝑡 = 0 you make contact with an enemy
destroyer. Immediately you start “porpoising” (varying your depth). Let the surface of the water be 𝑦 = 0. At time
𝑡 = 4 minutes you are at your deepest, 1000 meters below the surface. At time 𝑡 = 9 minutes you are at your
shallowest point, 200 meters below the surface. Your depth varies sinusoidally with time. Your submarine is safe
from the enemy destroyer when it is 300 meters or more below the surface. During what time interval is your
submarine first safe?
6. You are on the beach in Galveston. At 2:00 pm on June 15th the tide is in (the water level is at its deepest). At that
time you find that the water depth at the end of the pier is 1.5 meters. At 8:00 pm the same day when the tide is
out, you find that the depth of the water is 1.1 meters. Assume that the depth of the water varies sinusoidally with
time. What will the depth of the water be at 7:00 am on June 16th?
7. Create your own situation that could be modeled with a sinusoidal function. Solve your question.