Respuesta :
Answer:
a) [tex]k = 0.000124[/tex]
b) According to these data, the Shroud of Turin has around 760 years.
Step-by-step explanation:
The amount of carbon-14 is modeled by the following equation:
[tex]C(t) = C_{0}e^{-kt}[/tex]
In which [tex]C_{0}[/tex] is the initial amount and k is the rate of decrease.
(a) Find the value of the constant k in the differential equation.
Half-life of 5595 years.
So [tex]C(5595) = 0.5C_{0}[/tex]
[tex]C(t) = C_{0}e^{-kt}[/tex]
[tex]0.5C_{0} = C_{0}e^{-5595k}[/tex]
[tex]e^{-5595k} = 0.5[/tex]
Applying ln to both sides
[tex]-5595k = -0.69[/tex]
[tex]k = 0.000124[/tex]
b) In 1988 three teams of scientists found that the Shroud of Turin, which was reputed to be the burial cloth of Jesus, contained about 91 percent of the amount of carbon-14 contained in freshly made cloth of the same material. How old is the Shroud of Turin, according to these data?
This is t when [tex]C(t) = 0.91C_{0}[/tex]
[tex]C(t) = C_{0}e^{-kt}[/tex]
[tex]0.91C_{0} = C_{0}e^{-0.000124t}[/tex]
[tex]e^{-0.000124t} = 0.91[/tex]
Applying ln to both sides
[tex]-0.000124t = -0.094[/tex]
[tex]t = 760.57[/tex]
According to these data, the Shroud of Turin has around 760 years.
