Charcoal from the remains of the fire found in the Lascaux caves was analysed.

It was determined that $85.5 \%$ of the amount of carbon-14 had decomposed since the wood was alive.

The Formula for Radiocarbon Dating is used:

$t = -8060 \ln r$

where:

$t$ denotes the age in years of the sample of wood which is to be determined
$r$ denotes ratio of the quantity of carbon-14 remaining in the sample after time $t$ to the quantity of carbon-14 in the sample at the time of its death.

Thus:

 $\ds t$ $=$ $\ds -8060 \times \map \ln {1 - 0.855}$ $\ds$ $=$ $\ds -8060 \times \ln 0.145$ $\ds$ $=$ $\ds -8060 \times \paren {-1.9310}$ $\ds$ $=$ $\ds 15 \, 573$

Hence it is determined that the fire in the cave, and hence the dwellers therein, dates from approximately $15 \, 500$ years ago.

## Historical Note

The Lascaux caves were discovered on $12$ September $1940$.

The walls of the caves were found to be covered in artwork from thousands of years ago.

Within the cave were found the remains of a fire.

By using radiocarbon dating on the charcoal remains of that fire, it was possible to obtain an approximate date of when the caves were occupied.