Lesson 4.4: Modelling Growth and Decay

Introduction

In this lesson, we will explore the fascinating world of exponential growth and decay. Exponential functions are vital in understanding various phenomena in the real world, including population dynamics, financial investments, and natural decay processes. Our learning objectives are to:

Develop exponential models for population growth, investment returns, and depreciation.
Interpret the parameters of growth or decay models in context.
Compare the behavior of exponential functions with linear models.
Set up an appropriate exponential model for a given growth or decay situation.
Understand the contextual meaning of model parameters.

Hook

Imagine a scenario where a small town has a population of 1,000 people, and this population grows at an annual rate of 5%. How many people do you think will live in that town in 10 years? Would you guess only a few more or exponentially more? This lesson will guide you through finding the answer and understanding the implications of exponential growth.

Exponential Models for Growth

What is an Exponential Model?

An exponential model describes growth that occurs at a rate proportional to its current value. The general form of an exponential growth function can be expressed as:

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

Where:

$P(t)$ is the population at time $t$.
$P_0$ is the initial population at $t = 0$.
$e$ is Euler's number, approximately equal to 2.71828.
$r$ is the growth rate (as a decimal).
$t$ is the time in suitable units.

Example 1: Population Growth

Let's consider our earlier example about the town. Suppose the initial population $P_0$ is 1,000 and the growth rate $r$ is 0.05 (5%). What will the population be after 10 years?

Using the formula:

P(10) = $1000 \cdot$ e^{$0.05 \cdot 10$}

Calculating the exponent:

$0.05 \cdot 10 = 0.5$

Thus,

P(10) = $1000 \cdot$ e^{0.5} $\approx 1000$ $\cdot 1$.$64872 \approx 1648$.72

Rounding to the nearest whole number, the population will be approximately 1,649 people after 10 years.

Interpreting the Growth Rate

The parameter $r$ represents the growth rate. A positive $r$ indicates that the population is increasing, while a negative $r$ would indicate decay. Understanding these rates within context helps in decision-making regarding resource allocation, urban planning, and other socioeconomic factors.

Exponential Models for Investment

Understanding Investment Growth

Investments often grow exponentially as well, following a similar formula as population growth. The formula for compound interest is:

$A(t) = P_0 \left(1 + \frac{r}{n}$

$ight)^{nt}$

Where:

$A(t)$ is the amount of money accumulated after $t$ years, including interest.
$P_0$ is the principal amount (the initial investment).
$r$ is the annual interest rate (as a decimal).
$n$ is the number of times that interest is compounded per year.
$t$ is the time the money is invested for, in years.

Example 2: Investment Growth

Let’s say you invest $1,000 at an interest rate of 5% per annum, compounded annually ($n = 1$), for 10 years. We can calculate the future value of the investment:

$A(10) = 1000 \left(1 + 0.05$

$ight)^{1 \cdot 10}$

Calculating the equation:

$A(10) = 1000 \cdot (1.05)^{10}$

Using the property of exponents:

$(1.05)^{10} \approx 1.62889$

So,

A(10) $\approx 1000$ $\cdot 1$.$62889 \approx 1628$.89

Thus, the investment would grow to approximately $1,629 after 10 years.

Interpreting Investment Models

In this context, $P_0$ is the initial investment amount, and $r$ is the interest rate. Understanding these parameters allows investors to make informed decisions about their financial futures, including weighing the benefits of different investment options.

Exponential Models for Depreciation

Understanding Depreciation

Depreciation refers to the decrease in value of an asset over time, often modeled using an exponential decay function. The formula for exponential decay is:

$V(t) = V_0 e^{-rt}$

Where:

$V(t)$ is the value of the asset at time $t$.
$V_0$ is the initial value of the asset.
$r$ is the rate of depreciation (as a decimal).
$t$ is the time in suitable units.

Example 3: Asset Depreciation

Suppose you purchase a car for $20,000, and it depreciates at a rate of 15% per year, meaning $r = 0.15$. What will the value of the car be after 5 years?

Using the formula:

V(5) = $20000 \cdot$ e^{-$0.15 \cdot 5$}

Calculating the exponent:

$-0.15 \cdot 5 = -0.75$

Thus,

V(5) = $20000 \cdot$ e^{-0.75} $\approx 20000$ $\cdot 0$.$47237 \approx 9447$.40

So, the car's value after 5 years will be approximately $9,447.

Interpreting Depreciation Parameters

For depreciation models, $V_0$ represents the initial value, and $r$ indicates how fast the asset is losing its value over time. This knowledge helps businesses manage their assets effectively, including planning for future purchases or sales.

Comparing Exponential with Linear Models

Understanding the Difference

Linear models increase or decrease at a constant rate, while exponential models change at a rate proportional to their current value. An example of a linear growth model can be expressed as:

$L(t) = L_0 + rt$

Where:

$L(t)$ is the linear quantity at time $t$.
$L_0$ is the initial quantity.
$r$ is the constant growth or decay rate.

Example 4: Linear vs. Exponential Growth

Let’s take the same initial population example of 1,000 people but assume it grows linearly at a rate of 50 people per year. The population at year 10 would be:

L(10) = 1000 + $50 \cdot 10$ = 1000 + 500 = 1500

Now, comparing this to our earlier exponential growth calculation where we found the population to be approximately 1,649, we see that exponential growth outpaces linear growth.

Visual Comparison

Graphing both functions would highlight the differences clearly:

The linear function presents a straight line, while the exponential function curves upward more steeply over time. This stark contrast shows why looking at growth in real-world contexts necessitates understanding the underlying function type.

Conclusion

In this lesson, we have learned the fundamental concepts of modeling growth and decay with exponential functions. We explored population growth, investment returns, and asset depreciation, noting how the interpretations of the parameters provide vital context. We also discussed the differences between exponential and linear models, reinforcing the importance of choosing the right model based on the scenario.

Study Notes

Exponential growth is modeled by the function $P(t) = P_0 e^{rt}$.
Investment growth uses a formula for compound interest: $A(t) = P_$0 \left(1$ + $\frac{r}{n}

ight)^{nt}.

Exponential decay can be described with $V(t) = V_0 e^{-rt}$.
Linear models grow or decay at a constant rate: $L(t) = L_0 + rt$.
Understanding $r$ helps interpret the implications of growth or decay in real-world situations.