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
set your calculator to radians and find
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.
And how did you come up with the max & min
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
Add:
y = 100
Therefore, the expected population of the rabbit colony in the year 2020 is 100.