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

Answer:

The age of the spear head is of 1938 yers.

Step-by-step 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

$Q(t) = Q(0)e^{-rt}$

$0.5Q(0) = Q(0)e^{-5700r}$

$e^{-5700r} = 0.5$

$\ln{e^{-5700r}} = \ln{0.5}$

$-5700r = \ln{0.5}$

$5700r = -\ln{0.5}$

$r = -\frac{\ln{0.5}}{5700}$

$r = 0.00012160476$

So

$Q(t) = Q(0)e^{-0.00012160476t}$

We have to find:

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

$Q(t) = Q(0)e^{-0.00012160476t}$

$0.79Q(0) = Q(0)e^{-0.00012160476t}$

$e^{-0.00012160476t} = 0.79$

$\ln{e^{-0.00012160476t}} = \ln{0.79}$

$-0.00012160476t = \ln{0.79}$

$0.00012160476t = -\ln{0.79}$

$t = \frac{-\ln{0.79}}{0.00012160476}$

$t = 1938.43$

Rounding to the nearest year.

The age of the spear head is of 1938 yers.