Question

there are some nasty little bacteria hiding under that rotten banana peel in your bedroom. there are 20 bacteria now and they grow continuously at a rate of 5% every hour. how long will it take for the bacteria population to double?

Explanation:

Step1: Set up the continuous - growth formula

The formula for continuous growth is $P(t)=P_0e^{rt}$, where $P(t)$ is the population at time $t$, $P_0$ is the initial population, $r$ is the growth rate, and $t$ is the time. Here, $P_0 = 20$, $r=0.05$, and we want to find $t$ when $P(t) = 40$ (since the population doubles). So we have the equation $40 = 20e^{0.05t}$.

Step2: Simplify the equation

Divide both sides of the equation $40 = 20e^{0.05t}$ by 20. We get $2=e^{0.05t}$.

Step3: Take the natural logarithm of both sides

Taking the natural - logarithm of both sides of the equation $2 = e^{0.05t}$, we have $\ln(2)=\ln(e^{0.05t})$. Since $\ln(e^{x})=x$, the right - hand side simplifies to $0.05t$. So $\ln(2)=0.05t$.

Step4: Solve for $t$

Divide both sides of the equation $\ln(2)=0.05t$ by 0.05. We know that $\ln(2)\approx0.6931$, so $t=\frac{\ln(2)}{0.05}=\frac{0.6931}{0.05}=13.862$ hours.

Answer:

$13.862$ hours