Trigonometry 2-12Mathematical Modeling

1
Trigonometry 2-12 Mathematical Modeling
Ex1. 2 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 waters surface was a
sinusoidal function of time. When his stopwatch
read 4 seconds, the point was at its highest, 16
feet above the waters surface. The wheels
diameter was 18 feet, and it completed a
revolution every ten seconds.
2
at t 4 sec, d 16 ft (high point)
18 ft
16 ft
3
a) Sketch a graph of the sinusoid.
-The amplitude is 9 since the radius is 9 and
the point goes 9 above and 9 below the center.
-The axis is 7 since the center of the wheel is
7 feet above the surface of the water.
-The phase shift is 4 on cosine, since the
point reaches its high point at 4 seconds.
4
16
7
t
4
14
-2
5
b) Write the equation for the sinusoid.
A 9
D 4
C 7
c) How far above the surface was the point at 5
seconds, at 17 seconds?
d(5) 14.281 feet
d(17) 4.219 feet
6
d) When did the point first reach the
waters surface?

t 0.082 seconds
7
Ex2. 4 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 time t 5.1 years.
8
a) Sketch the graph of the sinusoid.
-The amplitude is 300 since the difference
between the max and min is 600.
-The period is 4.4 since the difference in time
from the min to the next max is 5.1 2.9 2.2.
-The axis is 500, halfway between the max/min.
-The phase shift is 5.1 on cosine since the max
is reach at t 5.1 years.
9
(No Transcript)
10
b) Write an equation expressing the number of
foxes as a function of time.
A 300
D 5.1
C 500
c) Predict the population when t 7 years
f(7) 227 foxes
11
d) Foxes are endangered when their population
drops below 300. Between what two values of t
were the foxes first endangered?

300
between t 2.311 and 3.489 years