Lesson 7.4

Exponential Growth and Decay Models

Archaeologists use Carbon-14 dating to determine the age of ancient fossils, and epidemiologists use growth models to predict the spread of viruses. Both rely on the same underlying mathematics of exponential change.

Introduction

In the previous lesson, we learned the "Continuous Variation" formula for money. In science, we simply rename the variables. Instead of Principal , we have an initial population . Instead of an interest rate , we have a growth/decay constant .

1

Prerequisite Connection

You must be comfortable using and solving basic exponential equations. We will also use logarithms conceptually to solve for (though a deep dive into logs is in Chapter 8).

2

Today's Increment

We apply the exponential model to solve real-world problems involving bacterial doubling times and radioactive half-lives.

3

Why This Matters

Calculus allows us to work backwards: if we know how fast something is changing (the derivative ), we can predict how much of it there will be in the future (the integral). These models are the first step in that predictive power.

Key Concepts

The Universal Growth/Decay Model

1

: The quantity at time .

2

: The initial quantity (at ).

3

: The rate constant.

  • If , it is Growth.
  • If , it is Decay.

Doubling Time (Growth)

The time it takes for a population to double. At this time , .

Half-Life (Decay)

The time it takes for a substance to reduce to half its initial amount. At , .

(Note: is negative, making negative)

Worked Examples

Example 1: Predicting Population Growth

Basic

A bacteria culture starts with 100 cells and doubles every 3 hours. How many bacteria will be present after 5 hours?

1

Find k using Doubling Time

Use the shortcut: .

2

Build the Model

3

Evaluate at t = 5


Answer

Approximately 317 bacteria.

Example 2: Carbon-14 Decay

Intermediate

Carbon-14 has a half-life of 5,730 years. If a fossil sample contains 10 grams of C-14, how much will remain after 2,000 years?

1

Find k

2

Evaluate at t = 2000



Answer

Approximately 7.85 grams remain.

Example 3: Dating a Fossil

Advanced

A fossil is found to contain 40% of its original Carbon-14. Using the same half-life (5,730 years), estimate the age of the fossil.

1

Set up the Equation

We don't know , but we know the final amount is .

2

Solve for t

Divide both sides by : .
Take the natural log (ln) of both sides (a skill we formally learn next chapter, but introduced here as a solving tool).

3

Calculate Final Answer

Answer

The fossil is approximately 7,570 years old.

Common Pitfalls

Neglecting the Negative Sign for Decay

When dealing with half-life or decay, your value MUST be negative. If you use a positive , your fossil will appear to gain Carbon-14 over time!

Mismatched Units

If a half-life is given in days, then your time must also be in days. Always check your units before calculating.

Real-World Application

Forensics: Time of Death

Forensic scientists use Newton's Law of Cooling to estimate the time of death. This is a modified exponential decay model: .

By measuring the body's temperature and the room's temperature, they can solve for in reverse to find how long ago the cooling process began.

Practice Quiz

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