Problem 35 - Applied Calculus

Question

The following formulas give the populations of four different towns, \(A, B, C,\) and \(D,\) with \(t\) in years from now. $$\begin{array}{rl} P_{A}=600 e^{0.08 t} & P_{B}=1000 e^{-0.02 t} \\ P_{C}=1200 e^{0.03 t} & P_{D}=900 e^{0.12 t} \end{array}$$ (a) Which town is growing fastest (that is, has the largest percentage growth rate)? (b) Which town is the largest now? (c) Are any of the towns decreasing in size? If so, which one(s)?

Step-by-Step Solution

Answer

(a) Town D is growing fastest. (b) Town C is the largest now. (c) Town B is decreasing.

Step 1: Identify Growth Rates

The formulas for population growth in towns are given in the form \( P = P_0 e^{rt} \), where \(r\) is the growth rate. From the population functions:- Town A: \( r = 0.08 \) (8% growth per year)- Town B: \( r = -0.02 \) (-2% growth per year)- Town C: \( r = 0.03 \) (3% growth per year)- Town D: \( r = 0.12 \) (12% growth per year)Compare the growth rates to determine which town is growing the fastest.

Step 2: Determine Initial Populations

Evaluate the population models at \( t = 0 \) to find the current size of each town.- For Town A: \( P_A = 600 e^{0.08 \times 0} = 600 \)- For Town B: \( P_B = 1000 e^{-0.02 \times 0} = 1000 \)- For Town C: \( P_C = 1200 e^{0.03 \times 0} = 1200 \)- For Town D: \( P_D = 900 e^{0.12 \times 0} = 900 \)The town with the highest initial population is the largest now.

Step 3: Check for Decreasing Populations

A negative growth rate indicates a decreasing population. From the given growth rates: - Town A: Growth rate is 0.08 (positive, increasing) - Town B: Growth rate is -0.02 (negative, decreasing) - Town C: Growth rate is 0.03 (positive, increasing) - Town D: Growth rate is 0.12 (positive, increasing) Identify any town with a negative rate to confirm decreasing population size.

Key Concepts

Population Modeling

Population modeling is an essential concept in understanding how groups grow or shrink over time. It's like creating a future snapshot of how big or small the population in different towns will be. In this exercise, the populations of four towns—A, B, C, and D—are modeled with exponential functions. These functions help us see how quickly a town is growing or shrinking based on its current size and growth rate.

When we model populations, we often use a formula of the form:

• \( P = P_0 e^{rt} \)

Here, \( P \) represents the future population, \( P_0 \) is the initial population, \( r \) is the growth rate, and \( t \) is the time in years from now.

Understanding population models allows us to predict future population sizes, which is crucial for planning resources, infrastructure, and services. For example, if a town is rapidly growing, it might need more schools or homes. With slow growth or shrinking, it might see a different kind of planning.

Growth Rate Analysis

Growth rate analysis helps us determine how quickly a population changes over time. It's like measuring the speed at which a town is getting bigger or smaller. By looking at the growth rate in our exercise, we can compare the growth of different towns easily.

In this scenario, the growth rates for towns are identified by simply looking at the equations:

• Town A: 8% per year (0.08)
• Town B: -2% per year (-0.02)
• Town C: 3% per year (0.03)
• Town D: 12% per year (0.12)

The growth rate (\( r \)) positively correlates with how fast a population grows. A higher positive growth rate means a town is expanding quickly. So, in our example, Town D grows the fastest with a 12% growth rate.

On the flip side, a negative growth rate reflects a declining population. Town B shows a negative rate, signaling that its population is shrinking by 2% annually. Recognizing these rates can help explain population trends and guide decision-making.

Exponential Functions

An exponential function is a mathematical way to represent rapid growth or decay. In population studies, it's particularly useful for illustrating how populations expand or decline over time, often not in a straight line but rather a curve that gets steeper or shallower.

In our study, the formula \( P = P_0 e^{rt} \) captures each town's population dynamic. It's called an exponential function due to the presence of the constant \( e \), Euler's number, approximately 2.71828, which characterizes the speeding-up behavior seen in growth scenarios.

• The component \( e^{rt} \) rapidly increases the value of \( P \) when \( r \) is positive and decreases when \( r \) is negative.
• This function demonstrates that small changes in \( r \) can lead to dramatic differences in population over time.
• In the exercise, varying \( r \) values showcase different scenarios—from rapid growth in Town D to shrinkage in Town B.

Comprehending exponential functions allows us to predict future outcomes more accurately and understand complex systems like population dynamics.