Given that carbon-14 decays at a constant rate in such a way that it reduces to 50\% in 5568 years, find the age of an old wooden piece in which the carbon is only 12.5\% of the original.
(a)15836 years
(b)16668 years
(c)16704 years
(d)17552 years
Answer
Answer (as printed): C
Explanation
Let T denote the required number of 5568 year-intervals. Then, $x\left(1-\frac{50}{100}\right)^{\mathrm{T}}=12.5 \%$ of $x \Rightarrow\left(1-\frac{50}{100}\right)^{\mathrm{T}}$ $$=\frac{x}{8} \times \frac{1}{x}=\frac{1}{8} \Rightarrow\left(\frac{1}{2}\right)^{\mathrm{T}}=\left(\frac{1}{2}\right)^{3} \Rightarrow \mathrm{T}=3 .$$