Question

A warren of rabbits has a population of 25. The population is increasing at a rate of 20% per year. How can you write an exponential growth function to find the monthly growth rate?

Answer 1

Answer:

(See explanation)

Step-by-step explanation:

The differential equation for a exponential growth is:

$\frac{dn}{dt} = k\cdot n$

The formula is rearranged and integrated afterwards:

$\frac{dn}{n} = k \cdot dt$

$\int\limits_{n_{o}}^{n} {\frac{dn}{n} } = k\cdot \int\limits_{0}^{t}\, dt$

$\ln \frac{n}{n_{o}} = k\cdot t$

$\frac{n}{n_{o}} = e^{ k\cdot t}$

$n = n_{o}\cdot e^{k\cdot t}$

The exponential growth function can be used to find the montly growth function as follows:

$\ln 1.2 = k\cdot 12$

$k = 0.015$

The coeffcients of the expression are, respectively:

$n_{o} = 25$

$k = 0.015$

The exponential growth function with a monthly growth rate is:

$n(t) = 25\cdot e^{0.015\cdot t}$, where t is measured in months.