Biologists find that populations of some kinds of predatory

Biologists find that populations of some kinds of predatory animals vary periodically with time. Assume that the population of foxes in a certain forest varies sinusoidally with time. Records started being kept at time t = 0 years. A minimum number of 200 foxes appeared when t = 3 years. The next maximum, 800 foxes occurred at t = 5 years. On the graph paper below, sketch the graph of this sinusoidal function. Find an equation expressing the number of foxes as a function of time. Predict the fox population when t = 7 years. Foxes are declared an endangered species when their population drops below 300. Between w hat two nonnegative values of t were the foxes first endangered?

Solution

a) Midline = (800 + 200)/2 = 500 ========> y = 500

Amplitude = 800 – 500 = 300

Half of period = min to max = 5 – 3 = 2 years

Period = 4/4 = 1 year

Phase shift = 5 – 4 = 1

b) cosine phase shift = 1.0

Now 2pi/k = 4 ======> k = pi/2

y = 300*cos[pi/2*(t - 1)] + 500

c) At t = 7,

y = 300*cos[pi/2*(7 - 1)] + 500 = 200 foxes