Question

Mouse populations can double in 8 months $\left(A=8A_0\right)$. How long will it take for a mouse

population to triple?

Short answer

It will approx take$12.7$months.

Step-by-step solution

Step 1. Given information

$$\begin{gathered} A=2A_0 \\ t=8months \end{gathered}$$

Step 2. Writing the concept.

An amount$A_0$that grows or decays at rate k for certain time t, will be given by

$A=A_0{e}^{kt}$

Step 3. Calculation

Put the values of respective variables in the given formula:

$$\begin{gathered} A=2A_0 \\ t=8 \\ 2A_0=A_0{e}^{8k} \\ {e}^{8k}=\frac{2A_0}{A_0} \\ {e}^{8k}=2 \\ Takingnatural\log bothsides, \\ \ln e^{8k}=\ln 2 \\ 8k.\ln e=\ln 2 \\ k=\frac{1}{8}\ln 2 \\ NowsubstituteA=3A_0 \\ 3A_0=A_0{e}^{t\frac{1}{8}\ln 2} \\ {e}^{t\frac{1}{8}\ln 2}=\frac{3A_0}{A_0} \\ {e}^{t\frac{1}{8}\ln 2}=3 \\ Taking\log bothsides \\ \ln e^{t\frac{1}{8}\ln 2}=\ln 3 \\ \left(t\frac{1}{8}\ln 2\right)\ln e=\ln 3 \\ t=\frac{\ln 3}{t{\displaystyle \frac{1}{8}}\ln 2} \\ t\approx 12.7 \end{gathered}$$