Radioactive decay

Problem

A radioactive sample has a half-life of $8$ years. What fraction of material remains after $24$ years? And after $12$ years?

Solution

The residual fraction after a time $t$ is $\left(\tfrac{1}{2}\right)^{t/\tau}$ with $\tau = 8$ years.

• After $24$ years: $t/\tau = 24/8 = 3$ half-lives, so $\left(\tfrac{1}{2}\right)^3 = \boxed{\tfrac{1}{8}}$ remains.
• After $12$ years: $t/\tau = 12/8 = 1{,}5$, so $\left(\tfrac{1}{2}\right)^{3/2} = \dfrac{1}{2\sqrt{2}} = \dfrac{\sqrt{2}}{4} \approx \boxed{0{,}354}$ remains.

Links

Topics: Exponential function
Concepts: Exponential decay · Half-life
Functions: Exponential function
Skills: Modelling · Using formulae
Exercise type: Modelling problem