Final answer:
The cosine function to model the population of foxes in the certain forest is f(t) = 200*cos((2π/3)(t - 2)) + 500.
Step-by-step explanation:
To write the cosine function to model the population of foxes in a certain forest, we need to determine the amplitude, period, phase shift, and vertical shift of the cosine function.
Given that a minimum of 300 foxes appeared at t = 2 years and the next maximum of 700 foxes occurred at t = 5 years, we can determine the amplitude and period.
The amplitude is half the difference between the maximum and minimum values of the function, so it is (700 - 300)/2 = 200. The period is the length of one complete cycle, so it is 5 - 2 = 3 years.
Since the population varies sinusoidally with time, we can use the general equation for a cosine function:
f(t) = A*cos(B(t - C)) + D
where A is the amplitude, B is the frequency (2π/period), C is the phase shift, and D is the vertical shift.
Using the values we found, the cosine function to model this situation is:
f(t) = 200*cos((2π/3)(t - 2)) + 500