Question

scientists can use the percentage of carbon - 14 in a fossils remains (relative to carbon - 12) to estimate the age of the fossil. this technique is based on the predictable decay of radiometric isotopes.
scientists know that the half - life of carbon - 14 is 57 thousand (or 5,700) years. the decay of carbon - 14 is shown on this graph.
part b - thinking critically
french scientists used carbon - 14 dating to determine the age of prehistoric wall paintings in the niaux caves. they determined that the paintings were made using natural dyes approximately 13 thousand years ago.
use your knowledge of carbon - 14 dating to determine which two statements are true. (remember that the half - life of carbon - 14 is 5.7 thousand years.)
in another 5.7 thousand years, the niaux cave paintings will contain half as much carbon - 14 as they currently do.
the niaux cave paintings currently contain less than 1/8th of the carbon - 14 they contained when originally painted
cave paintings that are 7.3 thousand years old would contain twice as much carbon - 14 as the niaux cave paintings.
if scientists discovered other cave paintings with less carbon - 14 than the niaux cave paintings, those paintings must be younger.
when the niaux paintings were 11 thousand years old, they contained approximately 75% of their original carbon - 14.

Explanation:

Step1: Analyze the carbon - 14 half - life concept

The half - life of carbon - 14 is 5700 years. After each half - life, the amount of carbon - 14 is halved.

Step2: Check each statement

• Statement 1: In another 5700 years, the Niaux cave paintings will contain half as much carbon - 14 as they currently do

Since the half - life of carbon - 14 is 5700 years, after one more half - life (5700 years), the amount of carbon - 14 will be halved. So this statement is true.

• Statement 2: The Niaux cave paintings currently contain less than 1/8th of the carbon - 14 they contained when originally painted

The paintings are 13000 years old. \(13000\div5700\approx2.28\) half - lives. After 1 half - life (\(t = 5700\)), the amount is \(\frac{1}{2}\). After 2 half - lives (\(t=11400\)), the amount is \(\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}\). After 3 half - lives (\(t = 17100\)), the amount is \(\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}=\frac{1}{8}\). Since \(11400\lt13000\lt17100\), the amount is between \(\frac{1}{4}\) and \(\frac{1}{8}\), so this statement is false.

• Statement 3: Cave paintings that are 7300 years old would contain twice as much carbon - 14 as the Niaux cave paintings

The Niaux paintings are 13000 years old. \(13000 - 7300=5700\) years difference (one half - life). If the Niaux paintings have \(x\) amount of carbon - 14, paintings 7300 years old (1 half - life less) would have \(2x\) amount. So this statement is true.

• Statement 4: If scientists discovered other cave paintings with less carbon - 14 than the Niaux cave paintings, those paintings must be younger

Less carbon - 14 means more half - lives have passed. More half - lives mean older age. So this statement is false.

• Statement 5: When the Niaux paintings were 11000 years old, they contained approximately 75% of their original carbon - 14

\(11000\div5700\approx1.93\) half - lives. After 1 half - life (\(t = 5700\)), the amount is \(\frac{1}{2}=50\%\). After 2 half - lives (\(t=11400\)), the amount is \(\frac{1}{4} = 25\%\). At \(t = 11000\) (close to 2 half - lives), the amount is not \(75\%\). So this statement is false.

Answer:

The Niaux cave paintings currently contain less than \(1/8\)th of the carbon - 14 they contained when originally painted: False; In another \(5700\) years, the Niaux cave paintings will contain half as much carbon - 14 as they currently do: True; Cave paintings that are \(7300\) years old would contain twice as much carbon - 14 as the Niaux cave paintings: True; If scientists discovered other cave paintings with less carbon - 14 than the Niaux cave paintings, those paintings must be younger: False; When the Niaux paintings were \(11\) thousand years old, they contained approximately \(75\%\) of their original carbon - 14: False. So the two true statements are: In another \(5700\) years, the Niaux cave paintings will contain half as much carbon - 14 as they currently do and Cave paintings that are \(7300\) years old would contain twice as much carbon - 14 as the Niaux cave paintings.