Precalcululs Honors
Sinusoidal Applications Worksheet 1
Name:
1. Ferris Wheel Problem As you ride the Ferris wheel, your distance from the
ground varies sinusoidally with time. When the last seat is filled and the Ferris
wheel starts, your seat is at the position shown in the figure below. Let t be the
number of seconds that have elapsed since the Ferris wheel started. You find that
it takes you 3 seconds to reach the top, 43 feet above ground, and that the wheel
makes a revolution once every 24 seconds. The diameter of the wheel is 40 feet.
a. Sketch a graph of this sinusoidal function.
b. What is the lowest you go as the Ferris
wheel turns?
c. Why is it reasonable for your answer to
part b to be greater than 0?
d. Find an equation of this sinusoid.
e. Predict your height above ground when
1
you have been riding for 4 seconds.
3
f. Find the first three times you are 18 feet above ground.
2. Bouncing Spring Problem A weight attached to 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.
a. Sketch a graph of this sinusoidal function.
b. Find an equation expressing distance from the floor in terms
of the number of seconds on the stopwatch.
c. How high above the floor is the spring when the stopwatch
reads 17.2 seconds?
d. How high was the spring when the stopwatch was started?
e. When is the first time when the weight is 59 cm above the
floor?
⎛π
⎞
t − 3)⎟ + 23 e) 41.794 feet f) 9.965, 20.035, 33.965
(
⎝ 12
⎠
Answers: 1b) 3 feet d) Possible: y = 20cos
⎜
⎛ 2π
⎞
x − 0.3)⎟ + 50 2c) 43.308 cm, 2d) 58.092 cm 2e) 0.084 seconds
(
⎝ 3
⎠
seconds 2b) y = 10cos
⎜
3. Tidal Wave Problem A tsunami is a high-speed deep ocean wave caused by an
underwater earthquake. The water first goes down from its normal level, then rises the
same amount above its normal level, and finally returns to its normal level. This takes
about 15 minutes to complete.
A tsunami approaches a pier in Honolulu that has a normal water depth of 9 meters. At
its height the tsunami has a height of 19 m.
a. Assuming that the depth of water varies sinusoidally with time as the tsunami
passes, predict the depth of the water 2 minutes, 4 minutes, and 12 minutes after
it first reaches the pier.
b. According to your model, what will the minimum depth of the water be?
c. Explain your answer to part b in terms of what will happen in the real world.
d. What is the first time interval where there will be no water at the pier?
e. The “wavelength” of a wave is the distance between two consecutive crests of
the wave. If a tsunami travels at 1200 kilometer per hour, what is its
wavelength?
4. Tarzan Problem Tarzan is swinging back and forth on his grapevine. As he swings, he
goes back and forth across the riverbank, going alternately over land and water. Jane
decides to mathematically model his motion and starts her stopwatch. Let t be the
number of seconds the stopwatch reads and let y be the number of meters that Tarzan is
from the riverbank. Assume y varies sinusoidally with t and that y is positive when
Tarzan is over water and negative when he is over land.
Jane finds that when the stopwatch reads 2 seconds, Tarzan is at one end of his swing, 23
meters over land. She finds that 5 seconds after she has started the stopwatch, Tarzan
reaches the other end of his swing 17 meters over water.
a. Sketch a graph of Tarzan’s sinusoidal
function.
b. Find an equation expressing Tarzan’s
distance from the riverbank in terms of
t.
c. How far is Tarzan from the riverbank
when the stopwatch reads 2.8 seconds?
6.3 seconds? 15 seconds? Determine if
Tarzan was over land or water at these
times.
d. Where was Tarzan when Jane started the
stopwatch?
e. When was the first time Tarzan was
directly over the riverbank?
Answers: 3a) 1.569 meters, -0.945 meters, 18.511 meters 3b) -1 meters 3d) between 2.673 and 4.827
⎛π
⎞
t − 2 )⎟ − 3 4c) 16.382 m over land, 1.158 m over water, 13 m
(
⎝3
⎠
minutes 3e) 300 km 4b) y = −20cos
⎜
over land 4d) 7 m over water 4e) 0.356 seconds