Question

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?

Answer 1

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\%`