On May 1, 2008, the forest service stocked a lake with

On May 1, 2008, the forest service stocked a lake with 136 bluegill fish. On October 1, five months later, they estimated the population of bluegill in the lake to be 197. Assume the population of bluegill increased exponentially.

For this population of bluegill, find:

the 5-month percent change:  

the 5-month growth factor:  

the 1-month growth factor:  

the 1-month percent change:   

Define a function f that gives the number of fish in the lake t months after May 1, 2008.

Define a function g that gives the number of fish in terms of n , the number of 5-month periods after May 1, 2008.

Use a graphing calculator and the functions you defined above to determine how many months after May 1 are required for the population to surpass 750 bluegill. Your answer should be a whole number of months.

Solution

let Po be the initial population

and P is the final population

then , P = Po e^kt

in 5 months the polpulation has increased from 136 to 197

197 = 136 e^5k

on solving we get

k = .0740

percentage change of 5 months = 7.4%

growth factor is .0740

growth factor of 1 month is plug t = 1 in the equation P = Po e^kt

██████ = ███████ ███^██

████ █████████████ ████ ███████

██ = █████████████

███████████ ███████████████ ███ █ ██████████ = ███████████

███████████████████ ██████████████ ████ ██ ████████████ = ████████████

█(███) = ███ ██^██████████ &#███████;&#███████; ██████████████ ██████████ ████████████ ████████████ ████ █████████ █████ ███████ ████████████ █ ███████████████

███ ███████████████ ██████ ██████████████

█████ █(██) = ██████

███████ = ██████ ███^███████████

█████ ████████████████ ███ ████████

███ = █████████

█████████████████████ , █████████████████████████ ████████████ █████ ██████████ ██████████████████████ ████████ ████████████████ █████████ █████████████████