Number of half-lives

Problem

In how many half-lives does the residual quantity drop below $1/1000$ of the initial amount?

Solution

After $n$ half-lives the fraction $\left(\tfrac{1}{2}\right)^n$ remains. We require it to be less than $1/1000$: $\left(\tfrac{1}{2}\right)^n < \frac{1}{1000} \iff 2^n > 1000.$ Since $2^9 = 512 < 1000$ and $2^{10} = 1024 > 1000$, the smallest integer satisfying the inequality is $\boxed{n = 10 \text{ half-lives}}.$

Links

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