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?

Feb 5, 2017

Yearly change: 300%

Approx monthly: 12.2%

Explanation:

For $f \left(t\right) = 5 {\left(4\right)}^{t}$ where $t$ is expressed in terms of years, we have the following increase ${\Delta}_{Y} f$ between years $Y + n + 1$ and $Y + n$:

${\Delta}_{Y} f = 5 {\left(4\right)}^{Y + n + 1} - 5 {\left(4\right)}^{Y + n}$

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:

• ${\left(1 + \Delta M\right)}^{12} {f}_{i} = \left(1 + \Delta P\right) {f}_{i}$,

then

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