MathematicsChapter 4: Differential Equations

Exponential Decay and Half-Life

For decay y=y0e^(-kt); half-life satisfies e^(-kT)=1/2.

Learning objectives

State the rule for exponential decay and half-life.
Solve a exponential decay and half-life problem step by step.
Check the result, restrictions, notation, and units.

Definition

For decay y=y0e⁽-kt); half-life satisfies e⁽-kT)=12.

Mathematical form

y=y0e⁽-kt)

k=ln2/5.

Where / symbols

Exponential decay and half-life: For decay y=y0e^(-kt); half-life satisfies e^(-kT)=1/2.

CONCEPT AND RULE

For decay y=y0e⁽-kt); half-life satisfies e⁽-kT)=12.

METHOD

For exponential decay and half-life, identify the given information and required result. Select the matching definition or formula, state its conditions, substitute carefully, simplify one line at a time, and check.

WORKED EXAMPLE

If half-life is 5 years, k=ln2/5.

COMMON MISTAKE

Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.

How to solve exponential decay and half-life

Identify the mathematical objects, givens, restrictions, and required result.
Choose and write the relevant definition, theorem, or formula.
Substitute with brackets and carry out one justified step per line.
Simplify, state the final answer clearly, and verify it in the original setting.

Worked example

Worked example:
If half-life is 5 years, k=ln2/5.

In simple words

Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.

Key points

Exponential decay and half-life
rule
worked example

Exam preparation

Exam Focus

Quick revision

Revision Zone

Review the key facts, formulas and checks before you move on.