Problem 31 - Calculus and its applications

Question

The population of Payton County was 5400 at the last census and decreasing at the rate of \(2.5 \%\) per year. The total population of the county after \(t\) years, \(P(t),\) is given by \(P(t)=5400(0.975)^{t}\) Find \(\frac{P(8)-P(5)}{8-5}\). What rate of change does this represent?

Step-by-Step Solution

Answer

The average rate of change is approximately -120.6 people per year.

Step 1: Understanding the Problem

We need to find the average rate of change of the population of Payton County from year 5 to year 8. This involves calculating the population at both times and finding the difference, then dividing by the time difference.

Step 2: Calculate P(5)

Substitute 5 for \(t\) in the formula \(P(t) = 5400(0.975)^t\) to find the population at year 5.\[ P(5) = 5400(0.975)^5 \]Calculating gives:\[ P(5) \ \approx 5400 \times 0.882 \approx 4762.8 \]

Step 3: Calculate P(8)

Substitute 8 for \(t\) in the formula \(P(t) = 5400(0.975)^t\) to find the population at year 8.\[ P(8) = 5400(0.975)^8 \]Calculating gives:\[ P(8) \approx 5400 \times 0.815 \approx 4401 \]

Step 4: Calculate the Difference P(8) - P(5)

We find the difference between the populations at the two times:\[ P(8) - P(5) = 4401 - 4762.8 = -361.8 \]

Step 5: Calculate the Average Rate of Change

The average rate of change from year 5 to year 8 is the difference in population divided by the difference in time.\[ \frac{P(8) - P(5)}{8 - 5} = \frac{-361.8}{3} = -120.6 \]

Step 6: Conclusion

The average rate of change in the population from year 5 to year 8 is approximately -120.6 people per year. This indicates the population is decreasing.

Key Concepts

Average Rate of Change

In mathematics, the average rate of change is a way to describe the speed at which a quantity changes over time. It is essentially what we call a "slope" when dealing with straight lines. However, this concept also applies to curves, like in the case of population change over time, where the quantity in question changes continuously rather than in a perfectly linear fashion.
To calculate the average rate of change between two points in time, you take the difference in the values of the function at these points and divide it by the difference in time.
The formula is written as:

• Average Rate of Change = \( \frac{f(b) - f(a)}{b - a} \)

This provides a simple way to measure change over an interval even if the change is not constant between those points. In Payton County’s population example, we calculate the average rate of change of the population over three years, indicating how much the population decreases each year on average between year 5 and year 8.

Population Dynamics

Population dynamics is the study of how populations change over time. This can involve the study of birth rates, death rates, immigration, and emigration, along with other ecological factors that influence population.
A key concept in population dynamics is how population sizes can increase, decrease, or remain stable over time. In our case, Payton County’s population is experiencing exponential decay, which means it decreases by a constant percentage each year.
This change is modeled by:

• \( P(t) = 5400(0.975)^t \)

Here:

• The 5400 represents the initial population.
• The factor 0.975 indicates a 2.5% yearly decrease, as each year the population retains 97.5% of the previous year's size.

Understanding this model helps us predict how future populations might look based on current trends.

Mathematical Modeling

Mathematical modeling is the process of representing real-world situations using mathematical formulas and expressions. It involves assumptions and approximations to simplify complex systems.
In the context of population dynamics, modeling allows us to use mathematical formulas to describe, predict, and understand population changes over time.
For instance, Payton County's decreasing population is represented with an exponential decay model:

• \( P(t) = 5400(0.975)^t \)

This model shows us how to project the county's population for any given year after the initial census. The accuracy of such models depends largely on the assumptions made and the constancy of factors like the annual rate of decrease.
By using such mathematical models, we can better prepare for resource allocation, urban planning, and even ecological impacts, underscoring the importance of mathematical modeling in understanding real-life scenarios.