Question

In the summer time fruit flies are the worst. One morning I woke up to 5 fruit flies in my kitchen. Three hours later there were 13 fruit flies. If the growth rate is continuous, at what rate are the flies increasing? Round answers to four decimal places.

Answer 1

We are asked to determine the continuous growth rate. To do that we will use the following function:

$P(t)=P_0e^{kt}$

Where "P0" is the initial population, "k" is the growth rate, and "t" is time. Replacing the initial population of 5 we get:

$P(t)=5e^{kt}$

Now we are told that the population is 13 when the time is 3 hours. Replacing we get:

$13=5e^{3k}$

Now we solve for "k". First, by dividing both sides by 5:

$\frac{13}{5}=e^{3k}$

Now we take natural logarithm to both sides:

$\ln (\frac{13}{5})=\ln (e^{3k})$

Now we use the following property of logarithms:

$\ln x^y=y\ln x$

Applying the property:

$\ln (\frac{13}{5})=3k\ln e$

We have that the value of ln(e) is 1, therefore:

$\ln (\frac{13}{5})=3k$

Now we divide both sides by 3:

$\frac{1}{3}\ln (\frac{13}{5})=k$

Solving the operation we get:

$0.319=k$

Therefore, the growth rate is 0.319.