# The function y=50cos(3x)+75 describes the population of a rabbit colony in the national park x years after 2000. What is the population of the colony expected to be in the year​ 2020?

## so our x = 20

50 cos(60) + 75
= appr 27 rabbits

Strange question. The max population would be 125 and the minimum would be 25 repeatedly during that time, and it would go from a max to a min in a period of (1/2)(2π/3) years or appr. every year.

## To find the population of the colony expected in the year 2020, we need to substitute the value of x as 2020 - 2000 = 20 into the equation y = 50cos(3x) + 75.

Let's calculate it step by step:

Step 1: Subtract 2000 from 2020 to find the number of years after 2000: 2020 - 2000 = 20

Step 2: Substitute the value of x into the equation: y = 50cos(3x) + 75 becomes y = 50cos(3 * 20) + 75

Step 3: Calculate 3 * 20: 3 * 20 = 60

Step 4: Substitute the result back into the equation: y = 50cos(60) + 75

Step 5: Calculate cos(60): cos(60) = 0.5

Step 6: Substitute the result back into the equation: y = 50 * 0.5 + 75

Step 7: Calculate 50 * 0.5: 50 * 0.5 = 25

Step 8: Calculate the final population: y = 25 + 75 = 100

Therefore, the population of the rabbit colony expected in the year 2020 is 100.

## To find the expected population of the rabbit colony in the year 2020, we need to substitute the value of x when it represents the number of years after 2000.

First, we should calculate the year 2020 in terms of x. Since x represents the number of years after 2000, we can subtract 2000 from 2020 to get the number of years after 2000:

2020 - 2000 = 20

So, x is equal to 20 in this case.

Next, substitute x = 20 into the equation y = 50cos(3x) + 75:

y = 50cos(3 * 20) + 75

Evaluate the cos(3 * 20) first:

y = 50cos(60) + 75

Now, calculate the value of cos(60) using a scientific calculator or by referencing a trigonometric table. The value of cos(60) is 0.5:

y = 50 * 0.5 + 75

Multiply:

y = 25 + 75