$$ 1 \leftarrow $$ Use the formula $$ t=\frac{\ln 2}{\mathrm{k}} $$ that gives the time for a population, with a growth rate k, to double, to answer the following questions. The growth model $$ \mathrm{A}=9 e^{0.005 t} $$ 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? $$ \square $$ years (Round to the nearest whole number.)

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Given:
- The growth model for the population, $$ A $$, of a country in millions, $$ t $$ years after 2003 is $$ A = 9e^{0.005t} $$.
- The formula for the time for a population to double with a growth rate $$ k $$ is $$ t = \frac{\ln 2}{k} $$.

a. To find the country's growth rate, we need to determine the value of $$ k $$ in the growth model $$ A = 9e^{0.005t} $$.

The growth rate $$ k $$ is the coefficient of $$ t $$ in the exponential function. In this case, the coefficient of $$ t $$ is 0.005.

Therefore, the country's growth rate is $$ 0.005 	imes 100\% = 0.5\% $$.

b. To find how long it will take the country to double its population, we can use the formula $$ t = \frac{\ln 2}{k} $$ and substitute the value of $$ k $$ we found in part a.

Substitute $$ k = 0.005 $$ into the formula:
$$ t = \frac{\ln 2}{0.005} $$

Now, we can calculate the value of $$ t $$ to find out how long it will take the country to double its population.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\ln{\left(2ight)}}{0.005}$$
- step1: Rewrite the expression:
  $$\frac{\ln{\left(2ight)}}{\frac{1}{200}}$$
- step2: Multiply by the reciprocal:
  $$\ln{\left(2ight)}	imes 200$$
- step3: Reorder the terms:
  $$200\ln{\left(2ight)}$$
It will take approximately 139 years for the country to double its population.
a. The country's growth rate is 0.5%. b. It will take approximately 139 years for the country to double its population.

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