Question

Egyptian papyrus with 63% of its original carbon-14 atoms

Answer 1

Their ages are:

Egyptian papyrus: 3811.10 years

Aboriginal charcoal: 12510.36 years

Mayan headdress: 1905.55 years

Neanderthal skull: 28997.51 years

Why?

If we need to calculate how old is something, we can use the equation for radioactive decay rate. It's possible using the C-14 half-life (5740 years) as reference.

We must remember, C-14 is used because all the living things take up the element from the atmosphere, when an organism dies, the amount of carbon starts to decay (slowly).

We can calculate the decay rate using the following equation:

Decay rate half life:

$N_{t}=N_{o}*e^{-l*t}$

Where,

$N_{o}$ is the initial number of atoms (undecayed)

$N_{t}$ is the the number of atoms at time (undecayed)

$l$  the decay rate

Now, isolating the decay rate of the formula, we have:

$N_{t}=N_{o}*e^{-l*t}\\\\ln(\frac{N_{t}}{N_{o}})=-l*t$

Also, we can get the value of the decay rate (half life), using the following formula:

$l=\frac{0.693}{5740}=1.207x10^{-4}$

Now, calculating, we have:

Egyptian papyrus: With 63% of its original carbon-14 atoms.

$ln(\frac{63}{100})=-1.207x10^{-4}*t\\\\-0.46=-1.207x10^{-4}*t\\\\t=\frac{-0.46}{-1.207x10^{-4}}=3811.10years$

Aboriginal charcoal: With 22% of its original carbon-14 atoms.

$ln(\frac{22}{100})=-1.207x10^{-4}*t\\\\\\-1.51=-1.207x10^{-4}*t\\\\t=\frac{-1.51}{-1.207x10^{-4}}=12510.36years$

Mayan headdress: With 79% of its original carbon-14 atoms.

$ln(\frac{79}{100})=-1.207x10^{-4}*t\\\\\\-0.23=-1.207x10^{-4}*t\\\\t=\frac{-0.23}{-1.207x10^{-4}}=1905.55years$

Neanderthal skull: With 3% of its original carbon-14 atoms.

$n(\frac{3}{100})=-1.207x10^{-4}*t\\\\\\-3.50=-1.207x10^{-4}*t\\\\t=\frac{-3.50}{-1.207x10^{-4}}=28997.51years$

Have a nice day!