Question

The population of a bee hive is growing with a monthly percentage rate compounded continuously. The population doubles in 3 months.

Find the monthly percentage rate according to the exponential growth function. Leave your answer in terms of ln.

Answer 1

Answer:

$r = \frac{\ln{2}}{3}$

Step-by-step explanation:

The exponential growth function is:

$P(t) = P(0)e^{rt}$

In which P(t) is the population after t months, P(0) is the initial population and r is the monthly percentage rate.

The population doubles in 3 months.

This means that $P(3) = 2P(0)$

We use this to find r.

$P(t) = P(0)e^{rt}$

$2P(0) = P(0)e^{3r}$

$e^{3r} = 2$

$\ln{e^{3r}} = \ln{2}$

$3r = \ln{2}$

$r = \frac{\ln{2}}{3}$

Answer 2

Monthly percentage rate according to the exponential growth function is $\frac{ln3}{3}$

To get the monthly percentage rate, we will use the exponential function as shown;

$P(t) =P(0)e^{rt}$

• P0 is the initial population

• If the initial population doubles in 3 months, the final population will be 3P0

Substitute the given parameters into the formula to get the rate "r"

$P(t) =P(0)e^{rt}\\3P0=P(0)e^{rt}\\3=e^{3r}\\ln3 = 3r\\r=\frac{ln3}{3}\\r = 0.366\\r = 36.6 \%$

Hence monthly percentage rate according to the exponential growth function is $\frac{ln3}{3}$

Learn more on exponential function here: https://brainly.com/question/19742435