Question

The number of rainbow smelt in Lake Michigan had an average rate of change of −19.76 per year between 1990 and 2000. The bloater fish population had an average rate of change of −92.57 per year during the same time. If the initial population of rainbow smelt was 227 and the initial population of bloater fish was 1,052, after how many years were the two populations equal?

The linear function that models the population of rainbow smelt is y1 = −19.76x + 227, where
x = the years since 1990 and y1 = the number of rainbow smelt.

The linear function that models the population of bloater fish is y2 = _______.

The linear equation that determines when the two populations were equal is
____________________.

The solution is x = ____ years.

Answer 1

The linear function that models the population of bloater fish is y2 = -92.57x + 1052, where x = the years since 1990 and y2 = the number of bloater fish.

The linear equation that determines when the two populations were equal is:
-19.76x + 227 = -92.57x + 1052

Solving for x:
72.81x = 825
x = 825 / 72.81
x ≈ 11.33

Therefore, the two populations were equal after approximately 11.33 years, which is around the end of the 11th year (since the populations were likely discrete values).