Lesson 5.18
Applications: Half-Life & Doubling Time
The capstone of Unit 5: apply all your exponential and logarithmic skills to real-world problems involving half-life (decay) and doubling time (growth).
Introduction
"How long until this substance is half gone?" or "When will the population double?" These questions tie together exponents, logs, and modeling from the entire unit.
Past Knowledge
Exponential models (5.4–5.5), solving with logs (5.16), continuous growth ().
Today's Goal
Set up and solve half-life and doubling-time models.
Future Success
These models reappear in precalculus, chemistry, biology, and finance.
Key Concepts
Half-Life (Decay)
= initial amount
= half-life (time to halve)
= elapsed time
Doubling Time (Growth)
= initial amount
= doubling time
= elapsed time
Finding the Time
To find , isolate the exponential, take of both sides, and solve. For half-life: .
Worked Examples
Example 1: Half-Life
DecayA radioactive substance has a half-life of 8 years. Starting with 200 g, how much remains after 20 years?
Apply the half-life formula
Calculate
About grams remain
Example 2: Finding Half-Life
SolvingA sample decays from 100 mg to 25 mg in 12 hours. Find the half-life.
Set up the equation
Divide and take ln
Solve
Half-life = hours
Example 3: Doubling Time
GrowthA city's population grows at 3% per year (continuous). How long to double?
Set up with
Take ln and solve
Doubling time ≈ years
Common Pitfalls
Confusing Rate and Half-Life
If given a rate, use . If given a half-life, use . Don't mix the formulas.
Units Mismatch
If the half-life is in hours but is asked in minutes, convert! Units on and (or ) must match.
Real-Life Applications
Carbon-14 dating uses half-life to determine the age of fossils. Medical dosing relies on half-life to predict when a drug leaves the body. Investment planning uses doubling time () — known as the Rule of 70 — to estimate when savings will double.
Practice Quiz
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