(M) Integrated Math 1 B (IS895) \#90179 | Mastery Assess It-1 Due: Tomorrow 13. The population of a small town can be modeled by the equation $$ f(t)=4800(1.02)^{t} $$, where t is the number of years since the collection of the population data began. Which statement is true? The population was 4800 when the collection of population data began. The population will reach a maximum of 4800 people. The population grew by 102 people each year since the collection of population data began. The population grows at a rate of $$ 102 \% $$ each year.

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UpStudy AI · Accepted Answer

To analyze the given population model $$ f(t) = 4800(1.02)^{t} $$, let's break down the statements one by one.

1. **The population was 4800 when the collection of population data began.**
   - When $$ t = 0 $$ (the beginning of the data collection), we can find the population:
     $$
     f(0) = 4800(1.02)^{0} = 4800 \times 1 = 4800
     $$
   - This statement is **true**.

2. **The population will reach a maximum of 4800 people.**
   - The function $$ f(t) = 4800(1.02)^{t} $$ is an exponential growth function. It does not have a maximum value; it will continue to grow indefinitely as $$ t $$ increases. Therefore, this statement is **false**.

3. **The population grew by 102 people each year since the collection of population data began.**
   - The growth factor is $$ 1.02 $$, which means the population increases by $$ 2\% $$ each year. To find the actual increase in population, we can calculate:
     $$
     \text{Increase in year 1} = f(1) - f(0) = 4800(1.02)^{1} - 4800(1.02)^{0} = 4800 \times 1.02 - 4800 = 4800 \times (1.02 - 1) = 4800 \times 0.02 = 96
     $$
   - The population does not grow by 102 people each year; it grows by 96 people in the first year. This statement is **false**.

4. **The population grows at a rate of $$ 102 \% $$ each year.**
   - The growth rate is actually $$ 2\% $$ per year, not $$ 102\% $$. The $$ 1.02 $$ in the equation indicates a $$ 2\% $$ increase. Therefore, this statement is **false**.

### Conclusion:
The only true statement is:
- **The population was 4800 when the collection of population data began.**
The population was 4800 when the collection of population data began.

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