Exponential Growth and Decay
Grade 11 · Math · Study Guide
Model real-world phenomena like population growth, radioactive decay, and compound interest using exponential functions.
Exponential growth occurs when a quantity increases by a constant percentage each period: f(t) = a * b^t where b > 1. Exponential decay occurs when 0 < b < 1.
The natural base form is f(t) = a * e^(kt). When k > 0 the function grows; when k < 0 it decays. The initial value a is the quantity at t = 0.
Half-life is the time for a decaying quantity to reach half its value: t_half = ln(2)/k. Doubling time for growing quantities: t_double = ln(2)/k.
| Model | Formula | Use Case |
|---|---|---|
| Growth (discrete) | f(t) = a(1 + r)^t | Population, investments |
| Decay (discrete) | f(t) = a(1 - r)^t | Depreciation, drug clearance |
| Growth (continuous) | f(t) = ae^(kt), k > 0 | Compound interest, bacteria |
| Decay (continuous) | f(t) = ae^(kt), k < 0 | Radioactive decay, cooling |
| Quantity | Formula |
|---|---|
| Half-life | t_(1/2) = ln(2) / k |
| Doubling time | t_2 = ln(2) / k |
| Compound interest (continuous) | A = Pe^(rt) |
Key terms
- Exponential growth
- A function f(t) = a*b^t with b > 1; the rate of change is proportional to the current value.
- Exponential decay
- A function f(t) = a*b^t with 0 < b < 1; the quantity shrinks by a constant percentage each period.
- Half-life
- The time required for a decaying quantity to reduce to half its current amount.
- Doubling time
- The time required for a growing quantity to double; t_double = ln(2)/k.
Practice questions
A population starts at 500 and grows at 4% per year. What is the population after 3 years (to the nearest hundredth)?
- 540.00
- 560.00
- 562.43
- 624.00
Show answer
f(3) = 500(1.04)^3 = 500 x 1.124864 = 562.43.
A substance decays according to f(t) = 200e^(-0.1t). What is the initial amount?
- 0.1
- e
- 200
- 20
Show answer
At t = 0: f(0) = 200e^0 = 200 x 1 = 200.
A radioactive element has a half-life of 10 years. What fraction remains after 30 years?
Show answer
30 years = 3 half-lives. (1/2)^3 = 1/8 remains.
Which function represents exponential decay?
- f(x) = 3 * 2^x
- f(x) = 5 * (0.8)^x
- f(x) = 2^x + 1
- f(x) = e^x
Show answer
Base 0.8 is between 0 and 1, so this is exponential decay.
Find the doubling time for a population with continuous growth rate k = 0.05.
Show answer
t_double = ln(2)/0.05 = 0.6931/0.05 = 13.86 years.
Fun facts
- Carbon-14 has a half-life of about 5,730 years, which lets archaeologists date organic materials up to roughly 50,000 years old.
- If a population grows at just 2% per year, it doubles in about 35 years, illustrating how even small growth rates compound dramatically over time.
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