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Trigonometry 2-12 Mathematical Modeling

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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

(high point)

18 ft

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.

16

7

t

4

14

-2

b)Write the equation for the sinusoid.

A: 9

C: 7

D: 4

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

surface?

…

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

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.

800

500

200

t

2.9

5.1

b)Write an equation expressing the number

of foxes as a function of time.

A: 300

C: 500

D: 5.1

c)Predict the population

when t = 7 years

f(7) = 227 foxes

800

500

200

t

2.9

5.1

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