Question

use the graph at right to answer questions: 3. how long is a half - life for carbon - 14? 4. if a sample originally had 120 atoms of carbon - 14, how many atoms will remain after 16,110 years? 5. if a sample known to be about 10,740 years old has 400 carbon - 14 atoms, how many atoms were in the sample when the organism died?

Explanation:

Step1: Identify half - life from graph

From the graph, the time when the amount of carbon - 14 reduces to 50% of the initial amount is the half - life. The half - life of carbon - 14 is 5370 years.

Step2: Calculate number of half - lives for 16110 years

The number of half - lives $n=\frac{16110}{5370}=3$.
If the initial number of atoms $N_0 = 120$, and the formula for radioactive decay is $N = N_0\times(\frac{1}{2})^n$. Substituting $N_0 = 120$ and $n = 3$ into the formula, we get $N=120\times(\frac{1}{2})^3=120\times\frac{1}{8}=15$.

Step3: Calculate initial number of atoms for a 10740 - year old sample

The number of half - lives for 10740 years is $n=\frac{10740}{5370}=2$.
If the current number of atoms $N = 400$, and using the formula $N = N_0\times(\frac{1}{2})^n$, we can solve for $N_0$. Rearranging the formula gives $N_0=\frac{N}{(\frac{1}{2})^n}$. Substituting $N = 400$ and $n = 2$ gives $N_0=\frac{400}{(\frac{1}{2})^2}=400\times4 = 1600$.

Answer:

1. 5370 years
2. 15 atoms
3. 1600 atoms