Download Real World Trig applications

Real World Trig applications
Integrated IV: High Dive
Name:________________________
Date:___________ Period: _______
I.
The Animal Population
The following formula gives an animal population, P, as a function of time, t (in months), where
t = 0 is January 1st, 1980.
P = 230 cos 30t  + 3800.
a.) What is the animal population on February 1st of 1981? Show the algebraic set up please…
b.) The animal population is considered endangered when the population levels drop below 3700.
Will the population described above ever become endangered? If so when? Algebra please…
II.
The Pendulum Problem
A pendulum swings back and forth in a continuous motion. A star Integrated IV student decides to
model mathematically the motion and starts her stopwatch. Let t be the number of seconds the stop
watch reads and let y be the number of inches horizontally the pendulum is from the rest position
(y = 0 ). Assume y varies periodically with t, and that y is positive when the pendulum is swinging
right, and negative when swinging left.
The student finds that when t = 1 sec. The pendulum
is at one end of its swing, where y = 8 inches. The
student waits and realizes that it takes 4 seconds
for the pendulum to return back to the y = 8 position.
t =1
y = -8
y=0
y=8
a. ) Sketch a graph of this periodic function over the interval 0 < t < 10. Be sure to label your axes.
b.) Find the following:
Amplitude
Period
________
________
Frequency
Midline
________
________
c.) Write an equation expressing the pendulum’s distance, y, from the rest position in terms of
time, t.
d.) Predict the pendulum’s distance, y, when:
t = 3.8 seconds
t = 6.5 seconds
t = 15 seconds
e.) Where was the pendulum ½ second after the stop watch started?
f.) Find the smallest positive value of t for which the pendulum passes over the rest position
(i.e. y = 0).
III.
Surf’s UP!
A Tsunami is a tidal wave caused by an earthquake beneath the sea. These can raise the sea level
more than 100 feet in height and can travel at great speeds. Suppose that a particular wave series
was created by an earthquake which caused an additional height of 50 ft with 30 minutes between
each crest. At t = 0, there is no wave and the height of the wave is zero.
a.) Sketch a graph of the change in height of the ocean as a function of time (2 cycles please).
Be sure to label your axis.
b.) Determine the period, midline, frequency and amplitude.
c.) Write an equation that represents the height of the wave h(t) at time t.
d.) Some nutty surfers think they can handle waves that are higher than 20 feet. Over what
time intervals can they surf? Algebra please…