Respuesta :
Answer:
The estimated age of the skull is 39118 years.
Step-by-step explanation:
The amount of the substance after t years is given by:
[tex]A(t) = A(0)(1-r)^t[/tex]
In which A(0) is the initial amount, and r is the decay rate.
The half-life of carbon-14 is about 5600 years.
This means that [tex]A(5600) = 0.5A(0)[/tex]. We use this to find r, or 1 - r, to replace in the equation. So
[tex]A(t) = A(0)(1-r)^t[/tex]
[tex]0.5A(0) = A(0)(1-r)^{5600}[/tex]
[tex](1-r)^{5600} = 0.5[/tex]
[tex]\sqrt[5600]{(1-r)^{5600}} = \sqrt[5600]{0.5}[/tex]
[tex]1 - r = (0.5)^{\frac{1}{5600}}[/tex]
[tex]1 - r = 0.9999[/tex]
So
[tex]A(t) = A(0)(0.9999)^t[/tex]
Only 2% of the original amount of carbon-14 remains in the burnt wood of the campfire.
This is t for which [tex]A(t) = 0.02A(0)[/tex]. So
[tex]A(t) = A(0)(0.9999)^t[/tex]
[tex]0.02A(0) = A(0)(0.9999)^t[/tex]
[tex](0.9999)^t = 0.02[/tex]
[tex]\log{(0.9999)^t} = \log{0.02}[/tex]
[tex]t\log{0.9999} = \log{0.02}[/tex]
[tex]t = \frac{\log{0.02}}{\log{0.9999}}[/tex]
[tex]t = 39118[/tex]
The estimated age of the skull is 39118 years.
