Question

How old is a mammoth’s tusk if 25 percent of the original C-14 remains in the sample, if the half-life of C-14 is 5730 years?

Answer 1

Hint: Mammoth’s tusk is the enamel containing certain elements. The half life is calculated for the decay of elements and is useful for dating elements. C-14 is the isotope of carbon.

Formula used: Number of half lives n = $\dfrac{t}{{{t}^{{}^{1}/{}_{2}}}}$

Complete answer: We have been given the composition of the tusk of a mammoth whose 25 percent is the remaining carbon which is original and the remaining 75 percent has decayed. Now we have to find the life of that mammoth’s tusk when the half life of C-14 is given as 5730 years.
We know that number of half lives n = $\dfrac{t}{{{t}^{{}^{1}/{}_{2}}}}$,
Therefore, the life, t = n${{t}^{{}^{1}/{}_{2}}}$
The amount remaining of the compound, here C-14 isotope after it has reached its half life is half the amount remaining. So we can write it as,
Amount remained = $\dfrac{original\,amount}{{{2}^{n}}}$ or
A = $\dfrac{{{A}_{0}}}{{{2}^{n}}}$
This can be arranged as, $\dfrac{{{A}_{0}}}{A}={{2}^{n}}$
Now, we consider the original amount to be 100 percent and the amount remained after decay to be 25 percent. So, we will obtain,
$\dfrac{100}{25}={{2}^{n}}$
4 = ${{2}^{n}}$
Therefore, n =2
Now, keeping n as 2 and half life as 5730 years in the formula, t = n${{t}^{{}^{1}/{}_{2}}}$
We have, t = 2 $\times $ 5730 years
So, t = 11,460 years.
Hence, the mammoth’s tusk is found to be 11,460 years old.

Note: The half-life or the decay of elements is generally calculated for isotopes, so carbon 14 isotope is used here. The dating of carbon and calculation of its half-life or full life is used in archaeological surveys.