Respuesta :
Answer:
The age of the pottery bowl is 10523 years.
Step-by-step explanation:
Amount of substance:
The amount of a substance after t years is given by the following equation:
[tex]A(t) = A(0)(1-r)^t[/tex]
In which A(0) is the initial amount and r is the decay rate.
Half-life of C14.
Researching, the half-life of c-14 is 5730 years.
This means that:
[tex]A(5730) = 0.5A(0)[/tex]
We use this to find r. So
[tex]A(t) = A(0)(1-r)^t[/tex]
[tex]0.5A(0) = A(0)(1-r)^{5730}[/tex]
[tex](1-r)^{5730} = 0.5[/tex]
[tex]\sqrt[5730]{(1-r)^{5730}} = \sqrt[5730]{0.5}[/tex]
[tex]1 - r = 0.5^{\frac{1}{5730}}[/tex]
[tex]1 - r = 0.99987903922[/tex]
So
[tex]A(t) = A(0)(0.99987903922)^t[/tex]
Age of the pottery bowl:
We have that:
[tex]A(t) = 0.28A(0)[/tex]
We have to find t. So
[tex]A(t) = A(0)(0.99987903922)^t[/tex]
[tex]0.28A(0) = A(0)(0.99987903922)^t[/tex]
[tex](0.99987903922)^t = 0.28[/tex]
[tex]\log{(0.99987903922)^t} = \log{0.28}[/tex]
[tex]t\log{0.99987903922} = \log{0.28}[/tex]
[tex]t = \frac{\log{0.28}}{\log{0.99987903922}}[/tex]
[tex]t = 10523.1[/tex]
So
The age of the pottery bowl is 10523 years.
