Question

In 2Use the formula t=kthat gives the time for a population, with a growth rate k, to double, to answer the following que0.0031The growth model A = 6 e describes the population, A, of a country in millions, t years after 2003.a. What is the country's growth rate?

Answer 1

We have an exponential function for the population A(t) and we have to find the growth rate.

`A=6e^(0.003t)`

This can be expressed as:

`(A(t+1)-A(t))/(A(t))=(A(t+1))/(A(t))-1`

For the exponential formula we get:

`\begin{gathered} (A(t+1))/(A(t))-1 \\ (6e^(0.003(t+1)))/(6e^(0.003t))-1 \\ e^(0.003(t+1-t))-1 \\ e^(0.003\cdot1)-1 \\ 1.003-1 \\ k=0.003 \end{gathered}`

The growth rate is 0.003 or 0.3%.

We can calculate the time needed to double the population dividing ln(2) by the growth rate:

`t=(\ln (2))/(k)\approx(0.693)/(0.003)\approx231`

Answer:

a) The growth rate is 0.003 or 0.3%

b) The number of years to duplicate the population is 231 years.