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The question in the image asks you to build an exponential function model for population growth. Here's the information given:

- Initial population: 11,000 organisms
- Growth rate: 12.5% per year
- The exponential model is of the form \( P = ab^t \), where:
  - \( P \) is the population at time \( t \) (in years),
  - \( a \) is the initial population,
  - \( b \) is the growth factor (1 + growth rate),
  - \( t \) is the number of years.

### Steps to build the equation:

1. The initial population \( a \) is 11,000.
2. The growth rate is 12.5%, which as a decimal is \( 0.125 \). The growth factor \( b \) is \( 1 + 0.125 = 1.125 \).
3. The formula becomes:
   \[
   P = 11,000 \times (1.125)^t
   \]

Thus, the exponential model for the population growth is:

\[
P = 11,000 \times (1.125)^t
\]

Would you like more details or have any further questions? 

Here are 5 related questions to expand your understanding:
1. How would the equation change if the growth rate were 15% instead?
2. What happens to the population if the growth rate were negative (i.e., population decline)?
3. How can you determine the population after 5 years using the equation?
4. How would you adjust the model if the population starts at a different value?
5. How can you solve for \( t \) if you want to know when the population will double?

**Tip**: When dealing with exponential growth, remember that even small changes in the growth rate can have a significant impact over time!

A population numbers 11,000 organisms initially and grows by 12.5% each year. Build an exponential model in the form P = ab^t.

Learn how to model population growth exponentially with an initial population of 11,000 and a growth rate of 12.5% per year. This detailed solution explains exponential functions and applies them to population growth scenarios. Ideal for high school students.

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