https://file.aiquickdraw.com/mathbot/file/1727038253995kox01jb5.jpg

The image provides a mathematical function that models the population of Americans age 55 and older as a percentage of the total population. The function is given as:

\[
f(t) = 10.72(0.9t + 10)^{0.3} \quad \text{for} \quad 0 \leq t \leq 20
\]

where \( t \) is the number of years since 2000 (i.e., \( t = 0 \) corresponds to the year 2000).

### Questions to answer:

1. **Rate of change in 2001**: At what rate was the percentage of Americans age 55 and older changing at the beginning of 2001? (Find \( f'(t) \) and evaluate at \( t = 1 \)).

2. **Rate of change in 2014**: At what rate was the percentage of Americans age 55 and older changing in 2014? (Evaluate \( f'(t) \) at \( t = 14 \)).

3. **Percentage of population in 2014**: What was the percentage of Americans age 55 and older in 2014? (Evaluate \( f(t) \) at \( t = 14 \)).

### Approach:
1. To calculate the rate of change, we need to find the derivative of the function \( f(t) \), denoted \( f'(t) \), using the chain rule.
2. Once we have the derivative \( f'(t) \), we will substitute \( t = 1 \) (for 2001) and \( t = 14 \) (for 2014) to compute the rate of change.
3. To find the percentage of Americans 55 and older in 2014, we simply evaluate \( f(t) \) at \( t = 14 \).

Let me proceed with these calculations.Here are the results based on the provided function:

1. **Rate of change in 2001**: The percentage of Americans age 55 and older was changing at a rate of approximately **0.5437% per year** at the beginning of 2001.

2. **Rate of change in 2014**: The percentage of Americans age 55 and older was changing at a rate of approximately **0.3263% per year** in 2014.

3. **Percentage of population in 2014**: In 2014, the percentage of Americans age 55 and older was approximately **27.32%** of the total population.

Would you like further details on the calculations, or have any questions?

Here are 5 related questions for further exploration:
1. How does the rate of change compare between 2001 and 2014?
2. What will the percentage be in 2020 based on this model?
3. How would the model change if we adjusted the parameters to reflect a higher aging population rate?
4. What is the percentage change in population from 2000 to 2014?
5. Can you derive a formula for the year when the percentage growth will stop increasing?

**Tip**: Understanding how the derivative of a function reflects the rate of change can help interpret various real-world models, like population growth or decline.

At what rate was the percentage of Americans age 55 and older changing at the beginning of 2001 and 2014? What was the percentage of the population age 55 and older in 2014?

Explore the rate of change in the percentage of Americans age 55 and older from 2001 to 2014. Using calculus, we calculate the rate of change by differentiating the population model f(t) = 10.72(0.9t + 10)^0.3. The results show how the population growth slowed over time, with specific calculations for 2001 and 2014.

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