Question:

The population of a small town has been experiencing exponential growth. In 2010, the population was 5,000, and by 2020, it had increased to 12,000.

1. Determine the initial population growth rate, assuming exponential growth.
2. Write the exponential growth model for this town's population.
3. Use the model to predict the population of the town in 2030.

Answer:

1. To find the initial population growth rate, we can use the formula for exponential growth:

   

   Where:

   • P(t) is the population at time t
   • P₀ is the initial population
   • k is the growth rate

   We can rearrange the formula to solve for k:

   

   Substituting the given values into the formula, we have:

   

   Simplifying and evaluating the expression, we find:

   

   Therefore, the initial population growth rate is approximately 0.0693 or 6.93%.

2. Now that we know the growth rate, we can write the exponential growth model. Using the formula:

   

   This equation represents the population of the town at any given time t.

3. To predict the population of the town in 2030, we substitute t = 2030 into the exponential growth model:

   

   Evaluating the expression, we find:

   

   Therefore, the population of the town in 2030 is approximately 9,410.

Hence, the exponential growth model for the town's population is given by P(t) = 5000 * e^(0.0693t), and the predicted population for the town in 2030 is approximately 9,410.