# The population of rabbits in East Fremont is 250 in September of 2004, and growing at a rate of 3.5% each month. If the rate of population growth remains constant, determine the month and year in which the rabbit population will reach 128,000?

Mar 24, 2017

In October of $2019$ rabbit population will reach $225 , 000$

#### Explanation:

Rabbit population in sept 2004 is ${P}_{i} = 250$

Rate of monthly population growth is r=3.5%

Final population after $n$ months is P_f=128000 ; n= ?

We know ${P}_{f} = {P}_{i} {\left(1 + \frac{r}{100}\right)}^{n} \mathmr{and} {P}_{f} / {P}_{i} = {\left(1 + \frac{r}{100}\right)}^{n}$

Taking log on both sides we get $\log \left({P}_{f}\right) - \log \left({P}_{i}\right) = n \log \left(1 + \frac{r}{100}\right) \mathmr{and} n = \frac{\log \left({P}_{f}\right) - \log \left({P}_{i}\right)}{\log} \left(1 + \frac{r}{100}\right) = \frac{\log \left(128000\right) - \log \left(250\right)}{\log} \left(1.035\right) = 181.34 \left(2 \mathrm{dp}\right) \therefore n \approx 181.34$ months $= 15$ years and $1.34$ month.

In October of $2019$ rabbit population will reach $225 , 000$ [Ans]