Let's tackle the given problem step-by-step to provide a detailed solution.
Given:
- The population of Toledo, Ohio was approximately 510,000 in the year 2000.
- The population is increasing at a rate of 4.8% per year.
### Part (a)
We need to write the exponential function that describes the total population, [tex]\( P(t) \)[/tex], as a function of [tex]\( t \)[/tex], the number of years since 2000.
The general form of an exponential growth function is:
[tex]\[
P(t) = P_0 e^{rt}
\][/tex]
where:
- [tex]\( P_0 \)[/tex] is the initial population.
- [tex]\( r \)[/tex] is the growth rate.
- [tex]\( t \)[/tex] is the time in years since the initial time.
For our problem:
- [tex]\( P_0 = 510,000 \)[/tex]
- [tex]\( r = 0.048 \)[/tex] (since 4.8% as a decimal is 0.048)
Thus, our exponential function becomes:
[tex]\[
P(t) = 510000 e^{0.048t}
\][/tex]
### Part (b)
Next, we need to determine the rate at which the population is increasing in [tex]\( t \)[/tex] years. This requires finding the derivative of the population function [tex]\( P(t) \)[/tex] with respect to [tex]\( t \)[/tex].
Recall that the derivative of [tex]\( P(t) = P_0 e^{rt} \)[/tex] with respect to [tex]\( t \)[/tex] is:
[tex]\[
P'(t) = P_0 r e^{rt}
\][/tex]
For our given function:
[tex]\[
P'(t) = 510000 \cdot 0.048 \cdot e^{0.048t}
\][/tex]
Simplifying the constant term gives:
[tex]\[
P'(t) = 24480 e^{0.048t}
\][/tex]
This represents the rate at which the population is increasing at any time [tex]\( t \)[/tex] in years since 2000.
### Part (c)
Finally, we need to determine the rate at which the population is increasing in the year 2025. To find this, we evaluate the derivative [tex]\( P'(t) \)[/tex] at [tex]\( t = 25 \)[/tex] (since 2025 - 2000 = 25).
Substitute [tex]\( t = 25 \)[/tex] into the derivative function:
[tex]\[
P'(25) = 24480 e^{0.048 \cdot 25}
\][/tex]
The exact expression evaluates to a numerical value that indicates the rate of population increase. Using the given exact evaluation, we have:
[tex]\[
P'(25) \approx 81237 \text{ people per year}
\][/tex]
Rounding to the nearest person, the population of Toledo, Ohio is increasing at a rate of approximately:
[tex]\[
81237 \text{ people per year in 2025}
\][/tex]
This concludes the detailed solution of the problem.
