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A humanoid skull is discovered near the remains of an ancient campfire. Archaeologists are convinced the skull is the same age as the original campfire. It is determined from the laboratory testing that only 1% of the original amount of carbon-14 remains in the burned wood taken from the campfire. It is known that carbon-14 decays at a rate proportional to the amount remaining and that carbon-14 decay 50% over 5700 years.

a) Formulate a difference equation model for carbon-14 dating.

b) Determine the age of the humanoid skull found near the remains of the ancient campfire.

I get my a answer is

 an+1 =0.5an

but for part b, anyone can share me idea?

Vivian Tio
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1 Answers1

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Carbon-14 decays by $50 \%$ each $5700$ years, what we want to find out is when it has decayed to just $1 \%$. We can do it by setting up this equation:

$$\Big(\frac{1}{2}\Big)^n = \frac{1}{100}$$

Which is just $50 \%$ removed each $n$, until we reach $1 \%$. Solve for $n$ to get:

$$n = \frac{\log{\tfrac{1}{100}}}{\log{\tfrac{1}{2}}}$$ $$n = 6.643856...$$

So we need $n$ amount of iterations, which each is $5700$ years, to get to $1 \%$ of carbon-14; multiply them together to get that the age of the skull is $5700n \approx 37870$ years old.

Sam Anderson
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