The population sizes of many animal species rise and fall over time. Suppose that the population size of a certain species can be modeled by the following function. p(t) = 2670 - 1270 cos 1.1t In this equation, pt represents the total population size, and t is the time in years.
Find the following. If necessary, round to the nearest hundredth.
Amplitude of p:
Frequency of p: cycles per year
Time for one full cycle of p : years
To find the amplitude of p, we need to determine the highest and lowest points of the function.
The general form of a cosine function is:
f(x) = a cos(bx + c) + d
In this case, the function is p(t) = 2670 - 1270 cos(1.1t).
The amplitude of a cosine function is given by the absolute value of the coefficient of the cosine term. In our case, the coefficient is -1270. Thus, the amplitude of p is 1270.
To find the frequency of p, we look at the coefficient of t in the cosine term. In this case, it is 1.1.
For a cosine function, the frequency (in cycles per unit) is equal to 2π divided by the coefficient of the cosine term. Therefore, the frequency of p is given by 2π/1.1 ≈ 5.73 cycles per year.
Lastly, to find the time for one full cycle of p, we divide 2π by the frequency. In this case, it would be 2π/5.73 ≈ 1.10 years. Therefore, one full cycle of p takes approximately 1.10 years.