Question

A small lake on the island contains two species of fish: hake and redfish. The hake are predators that eat the redfish. The fish population in the lake varies periodically with period 180 days. The number of hake varies between 500 and 1500 , and the number of redfish varies between 1000 and 3000. The hake reach their maximum population 30 days after the redfish have reached their maximum population in the cycle. Find cosine functions of the form y=a cos k(t-b)+c that model the hake and redfish populations in the lake.

Answer 1

To model the hake and redfish populations in the lake, you can use cosine functions of the form y=a cos k(t-b)+c. The hake population can be modeled by the function y = 500 cos((2π/180)(t-30)) + 1000, and the redfish population can be modeled by the function y = 1000 cos((2π/180)(t+30)) + 2000.

Step-by-step explanation:

To model the hake population in the lake, we can use the cosine function y = a cos(k(t-b)) + c. Given that the period of variation is 180 days, we know that k = 2π/180. The maximum population of hake occurs 30 days after the maximum population of redfish, so we can set b = 30. The minimum and maximum values of the hake population are 500 and 1500, so we can set c = (500 + 1500)/2 = 1000. Finally, we need to determine the amplitude a. Since the amplitude is half the difference between the minimum and maximum values, we can set a = (1500 - 500)/2 = 500. Therefore, the cosine function that models the hake population is y = 500 cos((2π/180)(t-30)) + 1000.

To model the redfish population, we can follow the same steps. The only difference is that the maximum population of redfish occurs 30 days before the maximum population of hake, so we set b = -30. Therefore, the cosine function that models the redfish population is y = 1000 cos((2π/180)(t+30)) + 2000.

Answer 2

The cosine functions representing the populations of hake (predators) and redfish in the lake are given by:

`\[y_{\text{hake}} = 500 \cos\left((2\pi)/(180)(t-30)\right) + 1000\]`

`\[y_{\text{redfish}} = 1000 \cos\left((2\pi)/(180)t\right) + 2000\]`

Step-by-step explanation:

The cosine functions are of the form
`\(y = a \cos(k(t - b)) + c\)`, where:

(a) is the amplitude,

(k) is the frequency,

(b) is the phase shift, and

(c) is the vertical shift.

For the hake population
`(\(y_{\text{hake}}\))`, the amplitude is 500 (the population varies between 500 and 1500), the frequency is
`\((2\pi)/(180)\)` (since the period is 180 days), the phase shift is 30 days (as hake reach their maximum population 30 days after redfish), and the vertical shift is 1000.

For the redfish population
`(\(y_{\text{redfish}}\))`, the amplitude is 1000 (the population varies between 1000 and 3000), the frequency is also
`\((2\pi)/(180)\)`, the phase shift is 0 days, and the vertical shift is 2000.

These functions capture the periodic nature of the fish populations, with the hake population peaking 30 days after the redfish population reaches its maximum. The amplitude reflects the range of population variation, the frequency determines the period, the phase shift accounts for the time lag, and the vertical shift adjusts for the baseline population.