PreCalc
posted by Jo .
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 t = 0 years. A minimum number, 200 foxes, occured when t = 2.9 years. The next maximum, 800 foxes, occurred at t = 5.1 years
I have to find
Foxes are declared to be an endangered species when their population drops below 300. Between what two nonnegative values of "t" were foxes first endangered.
I found one to be about
6.71101496 years i don't know if this is one of the first two how do I deal and how do I find the other one?

In an earlier reply to this same problem to you I had established the equation to be
F = 300sin(5pi/11)(t4) + 500
I had also shown that this equation satisfies all the data values you gave and it looks like you used it to get your value of 6.7 years.
But remember that the sine is negative in the III and IV quadrants, so I also got a value of t = 7.889 yrs.
Also recall that our period was 4.4, so subtracting 4.4 from any of our answers would produce another set of valid solutions.
6.71  4.4 = 2.31
7.889  4.4 = 3.49
let's test this
if t = 2.31, F = 300.3
if t = 3.49, F = 300.3
let's take a value between 2.31 and 3.49, how about t=3
F = 203 , which is less than 300
So the foxes were first endangered between the times of 2.3 years and 3.5 years.