Question

The half-life of carbon-14 is 5730 years. A bone is discovered which has 10 percent of the carbon-14 found in the bones of other living animals. How old is the bone?.

Answer 1

The bone discovered which has 10 percent of the carbon-14 found in the bones of other living animals is 19154 years old.

The process of carbon-14 decay can be modeled using the exponential decay formula:

`\[ N(t) = N_0 \cdot e^(-rt) \]`

Where:

`\(N(t)\)` is the amount of substance at time
`\(t\)`.

`\(N_0\)` is the initial amount of substance.

`\(r\)` is the decay rate constant.

`\(t\)` is time.

Given that the half-life of carbon-14 is 5730 years, we can determine the decay rate constant
`\(r\)` using the formula for exponential decay:

Now, we're given that the bone has 10% of the carbon-14 found in living animals. This means the remaining amount after decay is 10% of the initial amount
`(\(N_0\))`.

Let
`\(t\)` be the time passed.

`\[ N(t) = 0.1 \cdot N_0 = N_0 \cdot e^(-rt) \]`

Since we know the relationship between the remaining amount and the initial amount, we can set up an equation to solve for
`\(t\)`:

Therefore, the bone is approximately 19154 years old.

Answer 2

The bone is approximately 11,460 years old.

Step-by-step explanation:

The age of the bone can be calculated based on the amount of carbon-14 present compared to the amount found in living animals. Carbon-14 has a half-life of 5730 years. If the bone has 10 percent of the carbon-14 found in living animals, it means that 90 percent has decayed. Using the half-life, we can determine that this decay corresponds to approximately two half-lives. Therefore, the bone is approximately 11,460 years old.