Expert: Scotto - 12/8/2008

Question
Hey Sir,
I need like serious help with this question. Could you please tell the equation how to graph and stuff .Thanks

1.   Naturalists find  that the populations of some kinds of predatory animals vary periodically. Assume that the populations 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 t=5.1 years.
a.   Sketch a graph of this sinusoid.
b.   Write an equation expressing the number of foxes as a function of time, t.
c.   Predict the population when t=7 years
d.   Foxes are declared to be an endangered species when their population drops below 300. Between what 2 nonnegative values of t were foxes first endangered?
e.   Show on your graph that your answers to part(d) are correct.  

Answer
When the function is sinusoidal, that means we have a cos(x) function involved.  The minimum should be at 0° and the maximum should be at 180° (0 and π radians).  I will use the cos(x) curve since the maximum is at 0.  If desired, the sin(y) curve could also be used, but the maximum on this is at y=π/2.

The difference between the max time (t=5.1) and the min time (t=2.9) should be π radians.  Let A=5.1-2.9.  Note that the maximum on the cos(x) curve occurs at 0.  To keep values of t positive, add 2.9 to the x value.

The minimum and maximum for the cos(x) curve is -1 and 1.  These need to be converted to 200 and 800, so the cos(x) function needs to be multiplied by (800-200)/(1 - -1) = 300.  The average is (800+200)/2 = 500.
The curve is then 300cos(2.2(t-5.1)/(2π)) + 500.

a. To sketch the graph, just draw a cos(x) curve but instead of 0, π/2, π, 3π/2, and 2π on the x axis, solve 2.2(t-5.1)/(2π) equals each of these values to get the appropriate value for t.  For example, when x is 0 for the cos(x) curve, the value of t can be found by 2.2(t-5.1)/(2π)=0.  This is at t=5.1 .  When x is π/2, t is the solution to 2.2(t-5.1)/2π = π/2, so t is (π²/2.2) + 5.1.

b. The equation was already given in the paragraph write before a.

c. Put t=7 into this function.

d. Solve the equation 300cos(2.2(t-5.1)/(2π)) + 500 = 300 for the first two t values where t>5.1, since 5.1 is equivalant to t=0.

e. I can't draw a graph on the computer.