Half Interested or More Interesting

Learning Focus

Model half-life style decay with exponential functions and compare it to interest-style growth.

Big idea

If a quantity starts at $A$ and halves every $h$ units of time:

$f(t)=A\left(\tfrac12\right)^{t/h}$

• Half-life $h$: time for the quantity to drop to half.
• Every $h$ units multiplies by $\tfrac12$, so after $k$ half-lives you have $A\left(\tfrac12\right)^k$.
• Growth with interest uses $A(1+r)^t$ with $r>0$; decay uses a factor between $0$ and $1$.

Worked example

A medicine has $80$ mg with half-life $3$ hours.

• After 3 hr: $40$ mg
• After 6 hr: $20$ mg
• After 9 hr: $10$ mg

Model: $f(t)=80\left(\tfrac12\right)^{t/3}$. At $t=6$: $80\cdot\tfrac14=20$ ✓.

Takeaways

• Count how many half-lives have passed, then multiply by $\tfrac12$ that many times.
• Decay factor is between 0 and 1; growth factor is greater than 1.
• The exponent $t/h$ converts time into number of half-lives.