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Mom's Basement University

Exponential Growth and Decay

What You’ll Learn

Section titled “What You’ll Learn”

In this lesson you’ll learn how to recognize, write, and solve problems involving exponential growth and exponential decay.

The Concept

Section titled “The Concept”

Exponential functions have the form:

y=a⋅bxy = a \cdot b^x

Where a is the initial amount (when x = 0) and b is the growth or decay factor.

  • Exponential growth: b > 1 (quantity increases over time)
  • Exponential decay: 0 < b < 1 (quantity decreases over time)

The left graph shows exponential growth. The green curve starts at 500 and climbs faster and faster. The right graph shows exponential decay. The orange curve starts at 200 and drops quickly at first, then levels off toward zero. The gold dots mark the initial values.

Common real-world forms:

Growth:

y=a(1+r)ty = a(1 + r)^t

where r is the growth rate per period and t is time.

Decay:

y=a(1−r)ty = a(1 - r)^t

or for half-life problems:

y=a(12)t/hy = a\left(\frac{1}{2}\right)^{t/h}

where h is the half-life.

Worked Example

Section titled “Worked Example”

1. Bacteria growth

A population of 500 bacteria grows at 8% per hour. Write the model and find the population after 5 hours.

P(t)=500(1.08)tP(t) = 500(1.08)^t P(5)=500(1.08)5≈500×1.4693≈735 bacteriaP(5) = 500(1.08)^5 \approx 500 \times 1.4693 \approx 735 \text{ bacteria}

2. Radioactive decay (half-life)

A substance has a half-life of 10 years. Starting with 200 grams, how much remains after 30 years?

y=200(12)t/10y = 200\left(\frac{1}{2}\right)^{t/10} y=200(12)30/10=200(12)3=200×18=25 gramsy = 200\left(\frac{1}{2}\right)^{30/10} = 200\left(\frac{1}{2}\right)^3 = 200 \times \frac{1}{8} = 25 \text{ grams}

3. Car depreciation

A car bought for 25,000 dollars depreciates at 12% per year. Value after 4 years:

V(t)=25000(0.88)tV(t) = 25000(0.88)^t V(4)=25000(0.88)4≈25000×0.5997≈14,993 dollarsV(4) = 25000(0.88)^4 \approx 25000 \times 0.5997 \approx 14{,}993 \text{ dollars}

Real-World Application

Section titled “Real-World Application”

Exponential growth and decay appear in many important areas:

  • Population growth and spread of diseases
  • Compound interest and investment growth
  • Radioactive decay and carbon dating
  • Depreciation of cars, equipment, and assets
  • Cooling/heating of objects (Newton’s Law of Cooling)

Example: understanding exponential decay helps scientists determine the age of ancient artifacts using carbon-14 dating.

Quiz

Section titled “Quiz”
$P(t) = 200(1.06)^t$ represents:
A substance with half-life 8 years uses the model:
If a car depreciates at 15% per year, the decay factor is:
In $y = a \cdot b^t$, exponential decay occurs when:
An investment of $\$500$ earns $4\%$ interest compounded annually. What is the value after $3$ years?