Question

use the formula ( t = \frac{ln 2}{k} ) that gives the time for a population, with a growth rate ( k ), to double, to answer the following questions.
the growth model ( a = 9e^{0.004t} ) describes the population, ( a ), of a country in millions, ( t ) years after 2003.
...
a. what is the country’s growth rate?
( square% )
b. how long will it take the country to double its population?
years (round to the nearest whole number.)

Explanation:

Step1: Identify the growth rate from the model

The population growth model is given by \( A = 9e^{0.004t} \). For an exponential growth model \( A = A_0e^{kt} \), the growth rate \( k \) is the coefficient of \( t \) in the exponent. So here, \( k = 0.004 \). To convert this to a percentage, we multiply by 100: \( 0.004\times100 = 0.4\% \).

Step2: Calculate the doubling time

We use the formula \( t=\frac{\ln 2}{k} \). We know \( k = 0.004 \) and \( \ln 2\approx0.6931 \). Substitute these values into the formula: \( t=\frac{0.6931}{0.004} \).

Step3: Perform the division

\( \frac{0.6931}{0.004}=173.275 \). Rounding to the nearest whole number, we get \( t\approx173 \).

Answer:

a. The growth rate is \( 0.4\% \).
b. It will take approximately \(\boxed{173}\) years.