Question

An Archaeologist in Turkey discovers a spear head that contains 79% of its original amount of C-14. Find the age of the spear head to the nearest year.

Answer 1

The actual answer is 2357

Answer 2

The age of the spear head is of 1938 yers.

Explanation:

The amount of carbon 14 after years is given by the following equation:

`Q(t) = Q(0)e^(-rt)`

In which Q(0) is the initial amount and r is the decay rate.

The half life of carbon 14 is 5700 years.

This means that
`Q(5700) = 0.5Q(0)`

So

`Q(t) = Q(0)e^(-rt)`

`0.5Q(0) = Q(0)e^(-5700r)`

`e^(-5700r) = 0.5`

`\ln{e^(-5700r)} = ln(0.5)`

`-5700r = ln(0.5)`

`5700r = -ln(0.5)`

`r = -(ln(0.5))/(5700)`

`r = 0.00012160476`

So

`Q(t) = Q(0)e^(-0.00012160476t)`

We have to find:

t for which Q(t) = 0.79Q(0). So

`Q(t) = Q(0)e^(-0.00012160476t)`

`0.79Q(0) = Q(0)e^(-0.00012160476t)`

`e^(-0.00012160476t) = 0.79`

`\ln{e^(-0.00012160476t)} = ln(0.79)`

`-0.00012160476t = ln(0.79)`

`0.00012160476t = -ln(0.79)`

`t = (-ln(0.79))/(0.00012160476)`

`t = 1938.43`

Rounding to the nearest year.

The age of the spear head is of 1938 yers.