If a population is growing exponentially and continuously, how many years would it take the population to triple given that r = 0.07?
Understand the Problem
The question asks us to compute the number of years it takes for a population that is growing exponentially to triple in size, given a continuous growth rate of 0.07.
Answer
$t = \frac{\ln(3)}{0.07} \approx 15.69$ years
Answer for screen readers
It would take approximately $15.69$ years for the population to triple.
Steps to Solve
- State the exponential growth formula
The formula for continuous exponential 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 in years.
- Set up the equation
We want to find the time $t$ when the population triples, so $P(t) = 3P_0$. Substitute this into the formula: $3P_0 = P_0e^{rt}$
- Solve for t
Divide both sides by $P_0$: $3 = e^{rt}$ Substitute $r = 0.07$: $3 = e^{0.07t}$
- Take the natural logarithm of both sides
$\ln(3) = \ln(e^{0.07t})$ $\ln(3) = 0.07t$
- Isolate t and calculate
Divide both sides by $0.07$: $t = \frac{\ln(3)}{0.07}$ $t \approx \frac{1.0986}{0.07}$ $t \approx 15.69$
It would take approximately $15.69$ years for the population to triple.
More Information
The natural logarithm, denoted as $\ln(x)$, is the logarithm to the base $e$, where $e$ is an irrational number approximately equal to 2.71828. It's the inverse function of the exponential function $e^x$.
Tips
A common mistake is using the wrong formula or forgetting to use the natural logarithm to solve for the exponent. Also, rounding prematurely can affect the accuracy of the final answer.
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