To solve this, we'll use the formula for exponential growth:

$P(t) = P_0 e^{rt}$

Where:

• $P(t)$ is the population at time $t$,
• $P_0$ is the initial population,
• $r$ is the growth rate (in decimal form),
• $t$ is the time in years.

a. Exponential Function

Given:

• Initial population ($P_0$) = 500,000,
• Growth rate ($r$) = 5% = 0.05.

The function becomes:

$P(t) = 500,000 e^{0.05t}$

b. Rate of Increase

To find the rate of population increase at a given time, we differentiate $P(t)$ with respect to $t$:

$\frac{dP}{dt} = 500,000 \cdot 0.05 \cdot e^{0.05t}$

$\frac{dP}{dt} = 25,000 e^{0.05t}$

This represents the rate of increase of the population at any time $t$.

c. Rate of Increase in 10 Years

At $t = 10$, substitute $t = 10$ into the rate formula:

$\frac{dP}{dt} = 25,000 e^{0.05 \cdot 10}$

$\frac{dP}{dt} = 25,000 e^{0.5}$

Using $e^{0.5} \approx 1.64872$:

$\frac{dP}{dt} = 25,000 \cdot 1.64872$

$\frac{dP}{dt} \approx 41,218$

So, the rate at which the population is increasing in 10 years is approximately 41,218 people per year.

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Let me know if you want more details or further clarification. Here are some related questions to consider:

1. What would the population of Toledo be after 10 years?
2. How does the growth rate change when $r$ is reduced to 3%?
3. What would the population be in 20 years under this model?
4. How does the population growth rate compare after 5 years versus after 10 years?
5. What is the doubling time for this population?

Tip: Understanding how to differentiate exponential growth functions is crucial for solving population and economic growth problems.