# 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?

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]