Question

he radioactive element​ carbon-14 has a​ half-life of 5750 years. A scientist determined that the bones from a mastodon had lost 77.8 ​% of their​ carbon-14. How old were the bones at the time they were​ discovered?

Answer 1

The bones were 12,485 years old at the time they were​ discovered.

Explanation:

Amount of the element:

The amount of the element after t years is given by the following equation, considering the decay rate proportional to the amount present:

`A(t) = A(0)e^(-kt)`

In which A(0) is the initial amount and k is the decay rate, as a decimal.

The radioactive element​ carbon-14 has a​ half-life of 5750 years.

This means that
`A(5750) = 0.5A(0)`, and we use this to find k. So

`A(t) = A(0)e^(-kt)`

`0.5A(0) = A(0)e^(-5750k)`

`e^(-5750k) = 0.5`

`\ln{e^(-5750k)} = ln(0.5)`

`-5750k = ln(0.5)`

`k = -(ln(0.5))/(5750)`

`k = 0.00012054733`

So

`A(t) = A(0)e^(-0.00012054733t)`

A scientist determined that the bones from a mastodon had lost 77.8 ​% of their​ carbon-14. How old were the bones at the time they were​ discovered?

Had 100 - 77.8 = 22.2% remaining, so this is t for which:

`A(t) = 0.222A(0)`

Then

`0.222A(0) = A(0)e^(-0.00012054733t)`

`e^(-0.00012054733t) = 0.222`

`\ln{e^(-0.00012054733t)} = ln(0.222)`

`-0.00012054733t = ln(0.222)`

`t = -(ln(0.222))/(0.00012054733)`

`t = 12485`

The bones were 12,485 years old at the time they were​ discovered.