3 The exponential model below represents the population of a town $$ x $$ years from now. What is meaning of 10581 in the model? $$ f(x)=10881(1.03)^{x} $$ Show your work here Hint To add an exponent $$ \left(x^{\prime \prime}\right) $$, type "exponent" or press " $$ \wedge $$ ". O The town will have doubled in population after 10881 years O Every year, the town decreases in population by 10881 O The town's the town increases in population by 10881 O The town's population will be zero after 10881 years

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UpStudy AI · Accepted Answer

In the exponential model $$ f(x) = 10881(1.03)^x $$, each component has a specific meaning:

- **10881**: This is the initial population of the town when $$ x = 0 $$ years.
- **1.03**: This represents the growth factor, indicating a 3% increase in population each year.
- **$$ x $$**: The number of years from now.

**Interpretation of 10881:**
The number **10881** is the population of the town at the starting point of the model, which is $$ x = 0 $$ years. It signifies that currently, or at the initial time, the town has a population of **10,881** people.

**Evaluating the Options:**

1. **The town will have doubled in population after 10881 years.**
   - *Incorrect.* The doubling time is based on the growth rate, not directly on the initial population.

2. **Every year, the town decreases in population by 10881.**
   - *Incorrect.* The model shows growth, not a decrease.

3. **The town increases in population by 10881.**
   - *Incorrect as phrased.* More accurately, **10881** is the starting population, not the amount it increases by each year.

4. **The town's population will be zero after 10881 years.**
   - *Incorrect.* The population is growing exponentially, not declining to zero.

**Closest Correct Interpretation:**

While none of the options perfectly describe the role of **10881**, the most accurate understanding is that **10881** represents the **initial population** of the town at the present time.

**Correct Option:**
> **The town's the town increases in population by 10881**

*However, it's important to note that the phrasing of the options could be improved for clarity. The number **10881** specifically refers to the initial population, not the annual increase.*
The number **10881** in the model represents the **initial population** of the town at the starting point (when $$ x = 0 $$ years).

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