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Trigonometry 2-12 Mathematical Modeling

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					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
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)


16 ft   18 ft   1revolution
                10 seconds
  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.
                                          1
-The period is 10 since the frequency is    .
                                         10
-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
-2   4   14
b)   Write the equation for the sinusoid.
               2   
     A: 9   B:          C: 7   D: 4
               10   5
                            
            d(t)  7  9 cos (t  4)
                            5

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?
          
 7  9 cos (t  4)  0
          5
               …
                        5     1  7
                 t  4  cos   
                                 9
                        5
                    4  ( 2.4619  2n)
                        
                     7.9182  10n
                   
                     0.0818  10n

      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.
                 2     
     A: 300 B :            C: 500 D: 5.1
                4.4    2.2
                             
       f(t)  500  300 cos     (t  5.1)
                            2.2

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?
               
     300
500 800 cos       (t  5.1)  300
              2.2          …
                             2.2     1  200 
                   t  5.1      cos        
     500
                                        300 
                             2.2
                      5.1      ( 2.3005  2n)
                              
     300
                       6.711  4.4n
     200             
                        3.489  4.4n
           between t 2.9                t
                     = 2.311 and 3.489 years
                               5.1

				
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