Half-Life

High School

Definition

How long it takes an entity that's decaying exponentially to diminish by half. Half-life is usually used in physics to look at the stability of atoms or radioactive decay. It is constant over an exponentially decaying entity's lifetime.

Worked examples

\(A(t) = 100 \cdot \left(\frac{1}{2}\right)^{t/5}\)

If the half-life is 5 years, the amount halves every 5 years: \(A(5) = 50\), \(A(10) = 25\), \(A(15) = 12.5\).

\(50 = 200 \cdot \left(\frac{1}{2}\right)^{t/8}\)

Starting with 200 units and half-life 8 hours, solve for when 50 units remain: \(t = 16\) hours (two half-lives).

Common mistakes

\(A(t) = A_0 - \frac{A_0}{2} \cdot t\) → \(A(t) = A_0 \cdot \left(\frac{1}{2}\right)^{t/h}\) Half-life decay is exponential, not linear; the amount halves by proportion, not by a constant subtraction.
After two half-lives, nothing remains. → After two half-lives, \(\frac{1}{4}\) of the original amount remains. Each half-life cuts the current amount in half; it never reaches zero in finite time.
Half-life changes as the substance decays. → Half-life is constant throughout the decay process. The definition states half-life is constant; it does not depend on how much remains.

Where you'll use it next

You'll use half-life when modeling exponential decay in calculus, studying radioactive isotopes in chemistry and physics, analyzing drug concentration in biology, and solving real-world decay problems in applied math.

Found in 1 StudyPug lesson

Exponential Decay: Half-Life Formula

Grade 12 Math

Understand what half-life means, learn the half-life formula, and practice finding remaining amounts or the half-life itself from real data.

12th Grade12th