Problem 21 - Applied Calculus

Question

The following formulas give the populations of four different towns, \(A, B, C\), and \(D\), with \(t\) in years from now. $$ \begin{array}{cl} P_{A}=600 e^{0.09 t} & P_{B}=1000 e^{-0.02 t} \\ P_{C}=1200 e^{0.03 t} & P_{D}=900 e^{0.12 t} \end{array} $$ (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 the fastest. (b) Town C is the largest now. (c) Town B is decreasing in size.

Step 1: Identifying Growth Rates

To find which town is growing the fastest, we look at the exponent of the base of the exponential function, as that represents the growth rate. The given formulas are: \( P_A = 600 e^{0.09t} \), \( P_B = 1000 e^{-0.02t} \), \( P_C = 1200 e^{0.03t} \), and \( P_D = 900 e^{0.12t} \). The rates are 0.09 for town A, -0.02 for town B, 0.03 for town C, and 0.12 for town D. The largest growth rate is 0.12.

Step 2: Evaluating Current Population

To find the largest town currently, evaluate each function at \( t = 0 \) since this corresponds to the present time. \( P_A(0) = 600 \times e^{0.09 \times 0} = 600 \)\( P_B(0) = 1000 \times e^{-0.02 \times 0} = 1000 \)\( P_C(0) = 1200 \times e^{0.03 \times 0} = 1200 \)\( P_D(0) = 900 \times e^{0.12 \times 0} = 900 \)The largest current population is 1200 in town C.

Step 3: Checking for Decreasing Populations

A negative exponent indicates a decreasing population. From the growth rates: \( P_B = 1000 e^{-0.02t} \) has a negative rate (-0.02), implying town B's population is decreasing. All other exponents are positive, indicating growth.

Key Concepts

Population Modeling

Population modeling is a mathematical technique used to predict population changes over time. It helps to understand and forecast the dynamics of a population, whether it be human settlements, species in an ecosystem, or even cells in a biological process. The key aspect of population modeling is the use of equations to represent how populations change.

In the context of the exercise given, each town's population is modeled using an exponential equation of the form \( P(t) = P_0 e^{rt} \), where:

• \( P(t) \) is the population at time \( t \).
• \( P_0 \) is the initial population when \( t = 0 \).
• \( r \) is the growth rate, which can be positive (growth) or negative (decline).
• \( t \) is the elapsed time, typically measured in years.

By applying these formulas, we can evaluate and predict how the population will change, allowing towns to plan for infrastructure, resources, and policy adjustments. Understanding each part of the equation is essential for analyzing future scenarios accurately.

This kind of modeling is particularly useful in planning and development, helping decision-makers to anticipate growth trends and respond to potential challenges.

Growth Rate Analysis

Growth rate analysis focuses on determining how quickly a population is increasing or decreasing over time. It crucially hinges on identifying the growth rate \( r \) in exponential equations. This rate directly affects the future size of a population and informs planning decisions.

In the exercise, the growth rates for towns A, B, C, and D are extracted from the exponents of their respective equations:

• Town A: \( r = 0.09 \)
• Town B: \( r = -0.02 \)
• Town C: \( r = 0.03 \)
• Town D: \( r = 0.12 \)

A positive growth rate means the population is increasing, while a negative rate means it is decreasing. Thus, Town D, with a growth rate of 0.12, is growing the fastest, while Town B is shrinking because it has a growth rate of -0.02. Such analysis helps highlight critical areas needing attention, like resources for growth or strategies to curb decline.

Growth rate analysis is a key tool in demography, business, and policy-making, as it offers insights into future trends based on current dynamics. This proactive approach aids in preparing for challenges or opportunities presented by population changes.

Exponential Functions

Exponential functions are mathematical expressions that describe processes where growth or decay is multiplicative. They play a significant role in models that involve rapid changes over time, such as population growth, radioactive decay, or compound interest.

The general form of an exponential function is \( y = a e^{bt} \), where:

• \( y \) is the output or result at time \( t \).
• \( a \) is the initial amount or size at \( t = 0 \).
• \( b \) is the rate of growth (if positive) or decay (if negative).
• \( t \) represents time, which is often the independent variable.

In our exercise, the exponential functions are used to illustrate how populations change. Town A, for instance, has its population expressed as \( P_A = 600 e^{0.09t} \), which shows continuous growth due to the positive exponent. A negative exponent, as in Town B's \( P_B = 1000 e^{-0.02t} \), indicates a decay in population over time.

Understanding exponential functions is critical for interpreting how quantities evolve. In the context of population modeling, they help predict future population sizes, guiding strategic planning for resources, housing, and services. They show not only current trends but also provide foresight on potential developments.