Find an explicit formula for the beetle population after n weeks. Pn = After how many weeks will the beetle population reach 219? ____ weeks

Question

A population of beetles are growing according to a linear growth model. The initial population (week 0) is , and the population after 6 weeks is .

Find an explicit formula for the beetle population after  weeks.

 = 

  After how many weeks will the beetle population reach 219? 

____  weeks

Answer 1

Po=9e^0r= 9=a

P6=69=ae^6r where r = annual interest rate = .33948

69/9 =e^6r

ln(23/3) = 6r

r = ln(23/3)/6 = about .33948 = 33.948% per year growth rate in population

assuming they continuously reproduce

e^b= 23/3

b=ln(23/3)

219= 9e^rt

r= about 9.3615 years to reach 219

if for some reason the growth rate were linear instead of exponential

then population increases by 10 each year

216-9 = 207 reached in 207/10 = 20.7 years

that's a maximum bound for time

another more realistic model is Pt = 9(1+r)^t

where r is about 40.4% and t also = a little over 9 years

this might make sense if there's one mating season per year

with

n mating seasons per year would give Pt= 9(1+r/n)^nt