Question

Initially, there were only 197 weeds at a park. The weeds grew at a rate of 25% each week. The following function represents the weekly weed growth: f(x) = 197(1.25)x. Rewrite the function to show how quickly the weeds grow each day and calculate this rate as a percentage.

Answer 1

So the original function given is

\(r(x)=197(1.25)^x\)
To make this simpler when we manipulate, we will write it as:
\(r(x)=197(1+0.25)^x\)

Now since that's the rate for a week, the daily rate of increase is 25% distributed over 7 days, right? Hence the new rate is 0.25/7=0.036. And since that's for a day, the power would be 7 TIMES every time the rate is compounded.

\(r(x)=197(1+\frac{0.25}{7})^{7x}\)

Does that make sense? You might need to simplify what's in the base.

Answer 2

You might realize that the method used to find the new function is much similar to compound interest.

\(r(x)=a(1+\frac{r}{n})^{nt}\)
a=initial number
r=rate
n=number of times the rate is 'compounded' or applied
t=time