If a population is growing exponentially and continuously, how many years would it take the population to triple given that r = 0.07?

Steps to Solve

1. Write down the formula for exponential growth

The formula for continuous exponential growth is: $$ P(t) = P_0 e^{rt} $$ 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.

2. Set up the equation to represent the population tripling

We want to find the time $t$ when the population $P(t)$ is three times the initial population $P_0$. Therefore, we have:

$$ 3P_0 = P_0 e^{rt} $$

3. Substitute the given growth rate

We are given that the growth rate $r = 0.07$. Substitute this value into the equation:

$$ 3P_0 = P_0 e^{0.07t} $$

4. Solve for t

First, divide both sides of the equation by $P_0$: $$ 3 = e^{0.07t} $$

Next, take the natural logarithm of both sides: $$ \ln(3) = \ln(e^{0.07t}) $$

Using the property of logarithms, $\ln(e^x) = x$, we get: $$ \ln(3) = 0.07t $$

Now, divide by 0.07 to solve for $t$:

$$ t = \frac{\ln(3)}{0.07} $$

Calculate the value of $t$:

$$ t \approx \frac{1.0986}{0.07} \approx 15.69 $$