The function #f(t)=5(4)^t# represents the number of frogs in a pond after #t# years. What is the yearly percent change? the approximate monthly percent change?

1 Answer
Feb 5, 2017

Answer:

Yearly change: 300%

Approx monthly: 12.2%

Explanation:

For #f(t)=5(4)^t# where #t# is expressed in terms of years, we have the following increase #Delta_Y f# between years #Y+n + 1# and #Y +n#:

#Delta_Y f =5(4)^(Y+n+1) - 5(4)^(Y+n) #

This can be expressed as #Delta P#, a yearly percentage change, such that:

#Delta P =(5(4)^(Y+n+1) - 5(4)^(Y+n))/(5(4)^(Y+n)) = 4 - 1 = 3 equiv 300 \%#

We can then calculate this as an equivalent compounded monthly change, #Delta M#.

Because:

  • #(1+ Delta M)^(12) f_i= (1 + Delta P) f_i#,

then

  • #Delta M = (1+ Delta P)^(1/12) - 1 approx 12.2\% #