Need Help answering the Questions in the photo attached : Sine and cosine functi

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Need Help answering the Questions in the photo attached :

Sine and cosine functions are used primarily in physics and engineering to model oscillatory behavior, such as the motion of a pendulum or the current In an AC circuit, but these functions also arise m other sciences in this project, you will consider an application to biology - we use sine functions to model the populations of a predator and its prey. An isolated island is Inhabited by two species of mammals: lynx and hares. The lynx air predators who feed on the hares, their prey The lynx and hare populations changes cyclically, as graphed below in part A of the graph, the hares are abundant, so the lynx have plenty to eat and their population increases. By the time portrayed in part B, so many lynx are feed mg on the hares that the hares population declines. In part C, the hare population has declined so much that there h not enough food for the lynx, so the lynx population starts to decline. In part D, so many lynx have died that the hares have few enemies, and their population increases again. This takes us back to where we started, and the cycle repeats itself. The graph can be written using shifted sine curves of the form: y = a sin(b (t - c)) + d Find functions to the form y = a sin (b(t - c)) + d that model the lynx and hare populations. What are the functions? Graph both on your calculator- using the correct window so that it matches the graph on page 1 Add the lynx and hare populations to get a new function that models total mammal population on this island You do not need to simplify once you odd!' Graph this new function on your calculator. and record the result on graph paper Find the average value, amplitude, period and phase shift by approximating from your calculator Indicate these on your graph Write a single formula of the form y = a sin (b(t - c)) + d that models the total mammal population How are the average value and period of the total mammal population related to those of the two individual functions? The hare population equals 3000 at 4 different times during the 250 week period on the graph on page 1. Find these times. Start by using an algebraic method, then round off answers to 1 decimal place. Be sure to show your algebra method! In real life, most predator/prey populations do not behave as simply as the examples we have described here, in most cases, the populations of predators and prey oscillate, but the amplitudes get smaller and smaller, so that eventually both populations stabilize near a constant value. Modify the lynx and hare functions so that their amplitudes decrease exponentially by about 90% over 960 weeks. Record your functions and Sketch their graphs. Explain how you found your new amplitude functions. A small lake on an island contains two species of fish: hake and redfish. The hake are predators that eat the redfish. The fish population in the lake varies periodically every 180 days The number of hake varies between 500 and 1500, and the number of redfish vanes between 1000 and 3000. The hake reach their maximum population 30 days after the redfish have reached their maximum population in the cycle. Sketch a graph like the one on page 1 that shows two complete cycles of the population cycle for these species of fish. Assume that t-0 corresponds to a time when the redfish population is at a maximum. Find cosine functions of the form y = a cot(b(t - c))+d that model the hake and redfish populations.

Explanation / Answer

1) d is the average value of the function and it is average value of highest and lowest point in graph

For Hares: d = (3500 + 400)/2 = 3900/2 = 1950

For Lynx : d = (1900+400)/2 = 1150

Now period = 2pi/B

For Hares: Period = 120 = 2pi/B => B = 2pi/120 = pi/60

For Lynx : Period = 130 = 2pi/B => B = 2pi/130 = pi/65

Phase shift = -C/B

For Hares: Phase shift = 0 = -C/B => C =0

For Lynx : Phase shift = 0 = -C/B => C =0

Also Amplitude for Hares = A = 1950-400 = 1550

For Lynx : = A = 1150-400 = 750

So the sine function for Hares is given by :

y = 1550sin(pit/60-0)+1950 or y = 1550sin(pit/60)+1950

Sine function for Lynx is given by :

y = 750sin(pit/65-0) +1150 or y = 750sin(pit/65)+1150

2) y = 1150sin(pit/60)+1950 + 750sin(pit/65)+1150

=> y = 1150sin(pit/60)+750sin(pit/65)+3100

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