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.
In this equation, represents the total population size, and is the time in years.
p(t)=2670 - 1270 cos 1.1t
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 the function p(t), we can look at the coefficient in front of the cosine term. In this case, the coefficient is -1270. The amplitude is the absolute value of this coefficient, so the amplitude of p is 1270.
To find the frequency of the function p(t), we look at the coefficient of t in the argument of the cosine term. In this case, the coefficient is 1.1. The frequency is equal to 2π multiplied by the coefficient, so the frequency of p is 2π * 1.1.
To find the time for one full cycle of p, we take the reciprocal of the frequency. So the time for one full cycle of p is 1 / (2π * 1.1).
Let's calculate the values:
Amplitude of p: 1270
Frequency of p: 2π * 1.1
Time for one full cycle of p: 1 / (2π * 1.1)
To find the amplitude, frequency, and time for one full cycle of the function p(t) = 2670 - 1270 cos 1.1t, we need to understand the properties of the cosine function.
1. Amplitude (A):
The amplitude of a cosine function is the vertical distance from the midline to the highest or lowest point of the function. In this case, the amplitude is the coefficient of the cosine term, which is 1270. Therefore, the amplitude is 1270.
2. Frequency (f):
The frequency of a cosine function is the number of cycles it completes per unit of time. In this case, the coefficient of t in the cosine term is 1.1, which determines the speed or frequency of the oscillation. So, the frequency is 1.1 cycles per year.
3. Time for one full cycle (T):
The time for one full cycle is the period of the function, which is the time it takes for the cosine function to complete one full oscillation. The period can be found by dividing 2π by the frequency (2π/frequency). In this case, the frequency is 1.1 cycles per year, so the time for one full cycle is approximately 2π/1.1 = 5.72 years (rounded to the nearest hundredth).
Therefore, the answers are:
Amplitude of p: 1270
Frequency of p: 1.1 cycles per year
Time for one full cycle of p: 5.72 years (rounded to the nearest hundredth).
amplitude = 1270 beasties
cos 1.1 t = cos 2 pi f t = cos 2 pi t/T
so
2 pi f = 1.1
and
2 pi/T = 1.1