It was reported in the newspaper that a prehistoric village had been dated using radiocarbon dating.

The carbon-14 test had been used to measure the amount of radioactivity still present in the organic material found in the ruins.

It was determined that a settlement existed in that place as long ago as $7000 BCE$.

Given that:

the estimated half-life of carbon-14 used was $5600$ years
the newspaper report dated from about $1960$

the carbon-14 test must have shown that approximately $32 \%$ or $33 \%$ of carbon-14 was still present in the organic material at the time of discovery.

## Proof

From First-Order Reaction, we have:

$x = x_0 e^{-k t}$

where:

$x$ is the quantity of carbon-14 at time $t$
$x_0$ is the quantity of carbon-14 at time $t = 0$
$k$ is a positive number.

By definition of half-life, when $x = \dfrac {x_0} 2$, we have $t = 5600$.

So:

$e^{-5600 k} = \dfrac 1 2$

So:

$k = \dfrac {\ln 0.5} {-5600} = \dfrac {\ln 2} {5600}$

After $7000 + 1960$ years:

 $\ds \dfrac x {x_0}$ $=$ $\ds e^{-8960 \paren {\ln 2 / 5600} }$ $\ds$ $=$ $\ds 0.3299$

So there is between $32 \%$ and $33 \%$ remaining.

$\blacksquare$