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 water’s surface was a sinusoidal function of time. When his stopwatch
Trigonometry 2-12Mathematical 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 water’s surface was a
sinusoidal function of time. When his stopwatch
read 4 seconds, the point was at its highest, 16
feet above the water’s surface. The wheel’s
diameter was 18 feet, and it
completed a revolution
every ten seconds.
at t = 4 sec,
d = 16 ft
-The period is 10 since the frequency is .
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.
b)Write the equation for the sinusoid.
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
d)When did the point first reach the water’s
t = 0.082 seconds
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.
a)Sketch the graph of the
-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.
b)Write an equation expressing the number
of foxes as a function of time.
c)Predict the population
when t = 7 years
f(7) = 227 foxes
d)Foxes are endangered when their population
drops below 300. Between what two values
of t were the foxes first endangered?
between t = 2.311 and 3.489 years