# A population of rabbits follows the law of uninhibited growth. There are 5 rabbits initially and after 4 months there are 35. a.) How many rabbits will there be in one year? b.) How long will it take for the initial population to triple?

##### 1 Answer
Jun 25, 2017

a) Rabbit population after $12$ months is $1715$ and b) will tripple , i.e $15$ after $2.2583$ months.

#### Explanation:

The formula for uninhibited growth of rabbit is ${P}_{t} = {P}_{i} \cdot {e}^{k t}$ ,

where ${P}_{t} , {P}_{i} , k , t$ are population, initial population,

growth constant, and period in months.

P_4=35 , P_i=5,t=4 , k = ? :. 35 =5*e^(k*4) or e^(4k) =7 . Taking log on both sides, we get,

4k=ln7 ; [lne=1] or k =ln7/4 = 0.486478

a) k= 0.486478 , P_12=? :. P_12 = P_i*e^(kt)  or

${P}_{12} = 5 \cdot {e}^{0.486478 \cdot 12} = 1715$

b) P_t=15 ; t =? :. 15 = 5 * e^(0.486478*t)  or

${e}^{0.486478 \cdot t} = \frac{15}{5} = 3$ Taking log on both sides, we get,

$0.486478 \cdot t = \ln 3 \therefore t = \ln \frac{3}{0.486478} = 2.2583$ months

a) Rabbit population after $12$ months is $1715$ and b) will tripple ,

i.e $15$ after $2.2583$ months [Ans]