# 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 double?

Jul 25, 2016

The population of rabbits will double around the second half on May, 2006 or in the beginning of June, 2006, depending on when in September, 2004 we started our observation.

#### Explanation:

Growing at 3.5% a month is equivalent to multiplication by $1.035$ a month.

If this growth continues for $N$ months, the multiplication factor is ${1.035}^{N}$.

We have to determine $N$ if this multiplication factor equals (or exceeds for the first time) $2$.
It means, we have to solve an equation
${1.035}^{N} = 2$

The solution to this equation is $N = {\log}_{1.035} \left(2\right)$

According to the rules of operating on logarithms,
${\log}_{1.035} \left(2\right) = {\log}_{10} \frac{2}{\log} _ 10 \left(1.035\right) \approx \frac{0.301}{0.015} \approx 20.15$

Therefore, we can expect the population of rabbits to double during the 21st month.
If September, 2004 is month #0, the end of 20 months period falls on May, 2006. That means that the population of rabbits will double around the second half on May, 2006 or in the beginning of June, 2006, depending on when in September, 2004 we started our observation.