Question

An ancient skull has a carbon-14 decay rate of 0.75 disintegrations per minute per gram of carbon (0.75 dis/(min⋅gC) ). (Assume that living organisms have a carbon-14 decay rate of 15.3 dis/(min⋅gC) and that carbon-14 has a half-life of 5715 yr .)

How old is the skull?

Answer 1

The age of the skull can be estimated using the decay rate of carbon-14. By comparing the decay rate of the skull with that of living organisms, and knowing the half-life of carbon-14, we can calculate its age. The skull is estimated to be approximately 26,285 years old.

Step-by-step explanation:

The age of the skull can be estimated using the decay rate of carbon-14. The decay rate of the skull is given as 0.75 dis/(min⋅gC), while the decay rate of living organisms is 15.3 dis/(min⋅gC). Since carbon-14 has a half-life of 5715 years, we can calculate the age of the skull by comparing its decay rate with that of living organisms.

First, we calculate the ratio of the decay rates:

Ratio = Decay rate of skull / Decay rate of living organisms

Ratio = 0.75 dis/(min⋅gC) / 15.3 dis/(min⋅gC)

Ratio = 0.049

Next, we can determine the number of half-lives that have passed:

Number of half-lives = ln(Ratio) / ln(0.5)

Number of half-lives = ln(0.049) / ln(0.5)

Number of half-lives ≈ 4.60

Finally, we can calculate the age of the skull:

Age = Number of half-lives × Half-life

Age = 4.60 × 5715 years

Age ≈ 26,285 years

Answer 2

The ancient skull is approximately 9,150 years old.

Step-by-step explanation:

Carbon-14 dating relies on the decay of carbon-14 isotopes in organic material. The decay rate is expressed as disintegrations per minute per gram of carbon
`(dis/(min⋅gC))`. The given decay rate for the ancient skull is 0.75
`dis/(min⋅gC)`, while living organisms have a decay rate of 15.3
`dis/(min⋅gC)`. The ratio of the current decay rate to the initial decay rate can be used to determine the number of half-lives that have passed.

The formula for the number of half-lives
`(n)` is given by:

`\[ n = (ln((N_t)/(N_0)))/(-0.693) \]`

where
`\(N_t\)` is the final amount (decay rate of the skull),
`\(N_0\)` is the initial amount (decay rate of living organisms), and 0.693 is the natural logarithm of 2. Substituting the values:

`\[ n = (ln((0.75)/(15.3)))/(-0.693) \]`

Solving for
`\(n\)` gives the number of half-lives that have elapsed. Multiply this by the half-life of carbon-14 (5715 years) to find the age of the skull:

`\[ \text{Age} = n * \text{Half-life} \]`

`\[ \text{Age} = n * 5715 \]`

`\[ \text{Age} = \left( (ln((0.75)/(15.3)))/(-0.693) \right) * 5715 \]`

After calculating, the ancient skull is approximately 9,150 years old, providing insight into its historical context.