Respuesta :
Answer:
The bones were 12,485 years old at the time they were discovered.
Step-by-step 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:
[tex]A(t) = A(0)e^{-kt}[/tex]
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 [tex]A(5750) = 0.5A(0)[/tex], and we use this to find k. So
[tex]A(t) = A(0)e^{-kt}[/tex]
[tex]0.5A(0) = A(0)e^{-5750k}[/tex]
[tex]e^{-5750k} = 0.5[/tex]
[tex]\ln{e^{-5750k}} = \ln{0.5}[/tex]
[tex]-5750k = \ln{0.5}[/tex]
[tex]k = -\frac{\ln{0.5}}{5750}[/tex]
[tex]k = 0.00012054733[/tex]
So
[tex]A(t) = A(0)e^{-0.00012054733t}[/tex]
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:
[tex]A(t) = 0.222A(0)[/tex]
Then
[tex]0.222A(0) = A(0)e^{-0.00012054733t}[/tex]
[tex]e^{-0.00012054733t} = 0.222[/tex]
[tex]\ln{e^{-0.00012054733t}} = \ln{0.222}[/tex]
[tex]-0.00012054733t = \ln{0.222}[/tex]
[tex]t = -\frac{\ln{0.222}}{0.00012054733}[/tex]
[tex]t = 12485[/tex]
The bones were 12,485 years old at the time they were discovered.
