A cloth fragment is found in an ancient tomb.It contains 63% of the carbon-14 (half-life is 5700 years) that it is assumed to have had initially.How old is the cloth?
A) $\quad 10,600$ years
B) $\quad 8,200$ years
C) $\quad 9,000$ years
D) $\quad 8,600$ years
E) none of these

Question

A cloth fragment is found in an ancient tomb.It contains 63% of the carbon-14 (half-life is 5700 years) that it is assumed to have had initially.How old is the cloth? 
A) $\quad 10,600$ years
B) $\quad 8,200$ years
C) $\quad 9,000$ years
D) $\quad 8,600$ years
E) none of these

Quizplus · Accepted Answer

The amount of carbon-14 remaining after t years is given by:$$N(t) = N_0 (1/2)^{t/t_{1/2}}$$where $N_0$ is the initial amount of carbon-14, $t_{1/2}$ is the half-life, and t is the age of the cloth.We are given that $N(t) = 0.63N_0$, so we can solve for t:$$0.63N_0 = N_0 (1/2)^{t/5700}$$$$0.63 = (1/2)^{t/5700}$$$$t = 5700 \log_2(0.63) = 8600 \text{ years}$$

A Cloth Fragment Is Found in an Ancient Tomb 10,600\quad 10,60010,600

A Cloth Fragment Is Found in an Ancient Tomb $\quad 10,600$