Honors Trig/Calculus

							Honors Trig/Calculus                                                  Name_____________________________
    Sec. 6.5D
Trig modeling from given information--Sinusoids

1. Breathing process The rhythmic process of breathing consists of alternating periods of inhaling and
exhaling in a sinusoidal pattern. One complete cycle normally takes place every 5 seconds. If F(t) denotes the
air flow rate at time t (in liters per second) and if the maximum flow rate is 0.6 liter per second, find an
equation expressing the air flow rate as a function of time.

a) What is the period of this model?___________ That means “b” equals _________.

b) The maximum flow rate gives you the amplitude (since the minimum is zero).
   That means “a” equals _________.

c) Write a sine equation relating the flow rate (F) as a function of the time (t)._________________________

d) Make a rough sketch illustrating the problem situation. (Positive flow rate—inhaling; negative flow rate—
                                                                                                exhaling)
                               1




                 Flow rate
                (liters/sec)                 5             10
                                                     time (seconds)

                               -1




e) Write a cosine equation after you figure out the phase shift.______________________________________

f) Check both equations on a graphing calculator. Pick a reasonable WINDOW and use RADIAN mode.
2. Wolf population Naturalists find that the population of wolves varies sinusoidally with time on a
particular island. After 2.5 years of keeping records, a maximum number of wolves was recorded, 1100.
After 5.2 years, a minimum number of wolves, 300, was recorded.

a) Make a rough sketch illustrating the problem.        800




                                                        600




                                                        400


                                      Wolf population
                                                        200




                                                                               5                   10
                                                                                   time (yea rs)


b) Amplitude:                                                                 a = ___________

   Axis:                                                                      d = ___________

   Period:                                                                    b = ___________

   Phase shift for a cosine graph:                                            c = ___________

   Phase shift for a sine graph:                                              c = ___________

c) Write two different equations (sine/cosine) relating the number of wolves (W) as a function of time (t).

   _______________________________________              ________________________________________

d) Graph one of the equations on your calculator. (RADIAN mode—choose a reasonable WINDOW)
   Check a few values (TRACE or VALUE (under 2nd TRACE)) to make sure your model is correct.
   Turn that one off and check the other equation.

e) Predict the population of wolves…                            (Change X-max in WINDOW as needed)

   …after 7 years of keeping records: _____________

   …after 9 years of keeping records:____________

   …after 15 years of keeping records:_____________

   …after 56.5 years of keeping records:__________            (hmm…you’ve seen that answer before, why did it
                                                                   happen now?________________________)

f) What are the first 8 times (to the nearest hundredth of a year) after keeping records that there are exactly
   1000 wolves on the island?

  __________,__________,___________,___________,_____________,_________,__________,_________

   (Use Y2 = 1000 and use INTERSECT under 2nd TRACE (CALC). Is it possible to find all 8 answers
    without using INTERSECT 8 times???)
3. Ferris wheel A ferris wheel is 24 feet in diameter and the bottom sits 3 feet off the ground. The wheel
rotates at a rate of 5 rpm. Beulah starts at the bottom of the ferris wheel and her vertical height above the
ground is calculated as she goes around.

a) Convert 5 rpm into ft/sec: _____________________________________________________________

   The period is the length of time for one complete revolution (in seconds). Period = _____________

b) Make a rough sketch illustrating the problem situation.


                             25




                             20
             Ferris wheel
             height (feet)
                             15




                             10




                              5




                                            10               20   time (seconds)   30



c) Amplitude:                                                                  a = ___________

   Axis:                                                                       d = ___________

   Period:                                                                     b = ___________

   Phase shift for a cosine graph:                                             c = ___________

   Phase shift for a sine graph:                                               c = ___________

d) Write two different equations (sine/cosine) relating the height (H) as a function of time (t).

   _______________________________________               _______________________________________

e) Graph one of the equations on your calculator. (RADIAN mode—choose a reasonable WINDOW)
   Check a few values (TRACE or VALUE (under 2nd TRACE)) to make sure your model is correct.
   Turn that one off and check the other equation.

f) Find how high is Beulah above the ground…

   …after 27 seconds:_________          …after 46 seconds:____________                  …after one minute:________

   …after 1 min. 35 sec.:_________      …after an hour:____________          …after 2 hrs 22 min 45 sec:______

g) Beulah can see all of Ashland if she is more than 19 feet off the ground. Give the first time interval this
   happens: _________________.