Exponential Growth & Decay

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Recognize when a situation follows exponential rather than linear patterns.
  • Use exponential growth and decay formulas to model real scenarios.
  • Interpret the meaning of the base and exponent in context (initial amount, growth/decay factor, and time).

Key Ideas

A quantity changes exponentially when it grows or shrinks by the same percentage or factor in each equal time interval.

The general model is:

\[ y = a \cdot b^t \]

where:

  • \(a\) is the initial amount,
  • \(b\) is the growth or decay factor,
  • \(t\) is the number of time periods.

Interpretation:

  • If \(b > 1\), the model represents growth.
  • If \(0 < b < 1\), the model represents decay.

Typical applications include:

  • populations or bacteria counts
  • compound interest
  • depreciation
  • radioactive decay

Common Problem Types

1. Write exponential models

Identify the initial value and growth/decay factor.

2. Convert percent change to a factor

  • Growth: \(b = 1 + r\)
  • Decay: \(b = 1 - r\)
    (where \(r\) is the rate in decimal form)

3. Evaluate the model at specific times

Compute \(a \cdot b^t\).

4. Interpret values in context

Explain what the model predicts and what the exponent means.

Strategies

  • Always convert percent changes into decimals before forming the factor.
  • Identify clearly what \(t\) represents (years, hours, days, etc.).
  • Use powers carefully; exponential change means multiplying, not adding.
  • For decay, think: “What percent is kept each cycle?”
  • Evaluate expressions like \(b^t\) separately to avoid common arithmetic mistakes.

Worked Examples

Example 1 — Simple Growth

A bacteria population doubles every hour. Initially there are 500 bacteria.

(a) Write a formula for the population after \(t\) hours.
(b) Find the population after 3 hours.

Solution:

  • Initial amount: \(a = 500\).
  • Doubling every hour → growth factor \(b = 2\).

Model:

\[ P(t) = 500 \cdot 2^t \]

At \(t = 3\):

\[ P(3) = 500 \cdot 2^3 = 500 \cdot 8 = 4000 \]

Example 2 — Simple Decay

A car loses 20% of its value each year. Its initial value is $25{,}000.

(a) Write a function for its value after \(t\) years.
(b) Find the value after 2 years.

Solution:

  • Losing 20% → keeping 80% → factor \(b = 0.80\).
  • Initial amount: \(a = 25{,}000\).

Model:

\[ V(t) = 25{,}000 \cdot 0.8^t \]

After 2 years:

\[ V(2) = 25{,}000 \cdot 0.8^2 = 25{,}000 \cdot 0.64 = 16{,}000 \]

WarningCommon Mistakes
  • Using the percent amount instead of converting to a factor (e.g., \(1+20\) instead of \(1+0.20\)).
  • Treating exponential situations as linear ones.
  • Misplacing the exponent (e.g., \(b \cdot t\) instead of \(b^t\)).
  • Forgetting that \(0 < b < 1\) indicates decay.

Practice Problems

  1. A population of 1,000 fish grows by 5% each year. Write a formula for \(P(t)\) after \(t\) years.
  2. Using your formula from (1), find \(P(3)\) (round to the nearest whole number).
  3. A radioactive substance loses 10% of its mass each hour. Initially, there are 80 grams. Write a function for the mass after \(t\) hours.
  4. Using your function from (3), find the mass after 2 hours.
  5. Decide whether the following describes exponential growth or decay:
    • A savings account earns 3% interest each year with no withdrawals.

1.
5% growth → factor
\[b = 1 + 0.05 = 1.05\]
Initial amount \(a = 1000\)

Model:
\[P(t) = 1000 \cdot 1.05^t\]

2.
\[P(3) = 1000 \cdot 1.05^3\]
Compute:
- \(1.05^2 = 1.1025\)
- \(1.05^3 \approx 1.157625\)

So:
\[P(3) \approx 1000 \cdot 1.157625 \approx 1158\]
Population ≈ 1,158 fish

3.
10% loss → keeps 90%:
\[b = 0.90\]
Initial mass = 80

Model:
\[M(t) = 80 \cdot 0.9^t\]

4.
\[M(2) = 80 \cdot 0.9^2 = 80 \cdot 0.81 = 64.8\]
Mass after 2 hours ≈ 64.8 grams

5.
Interest of 3% yearly → factor \(1.03\) each year →
This is exponential growth.

Summary

  • Exponential models multiply by the same factor each time period.
  • Growth uses \(b > 1\); decay uses \(0 < b < 1\).
  • The model \(y = a \cdot b^t\) describes many real processes.
  • Percent changes must be converted into decimal factors.
  • Interpreting results in context is essential.
  • Growth: \(b = 1 + r\); Decay: \(b = 1 - r\).
  • Exponential = repeated multiplication, not addition.
  • Identify the initial value and what \(t\) represents.
  • Check your factor: decay factors must be between 0 and 1.