Respuesta :
Using an exponential function, it is found that the Tequesta settlement is 1845 years old.
What is an exponential function?
The exponential equation for a decaying amount of a substance is given by:
[tex]A(t) = A(0)e^{-rt}[/tex]
In which:
- A(0) is the initial value.
- r is the decay rate, as a decimal.
Researching on the internet, the half-life of carbon 14 is of 5,730 years, hence A(5730) = 0.5A(0), which we use to find r.
[tex]A(t) = A(0)e^{-rt}[/tex]
[tex]0.5A(0) = A(0)e^{-5730r}[/tex]
[tex]e^{-5730r} = 0.5[/tex]
[tex]\ln{e^{-5730r}} = \ln{0.5}[/tex]
[tex]5730r = -\ln{0.5}[/tex]
[tex]r = -\frac{\ln{0.5}}{5730}[/tex]
[tex]r = 0.00012096809[/tex]
Hence, the equation is:
[tex]A(t) = A(0)e^{-0.00012096809t}[/tex]
The wood chips were found to contain 80% of the atmospheric carbon-14, hence we have to find t for which A(t) = 0.8A(0).
[tex]A(t) = A(0)e^{-0.00012096809t}[/tex]
[tex]0.8A(0) = A(0)e^{-0.00012096809t}[/tex]
[tex]e^{-0.00012096809t} = 0.8[/tex]
[tex]\ln{e^{-0.00012096809t}} = \ln{0.8}[/tex]
[tex]-0.00012096809t = \ln{0.8}[/tex]
[tex]t = -\frac{\ln{0.8}}{0.00012096809}[/tex]
[tex]t = 1845[/tex]
The Tequesta settlement is 1845 years old.
You can learn more about exponential functions at https://brainly.com/question/25537936
