Answer :
Answer:
The watch is 40.9 years old.
Explanation:
To know how many years old is the watch we need to use the following equation:
[tex] I_{(t)} = I_{0}e^{-\lambda t} [/tex] (1)
Where:
[tex]I_{(t)}[/tex]: is the brightness in a time t = (1/10)I₀
[tex]I_{0}[/tex]: is the initial brightness
λ: is the decay constant of tritium
The decay constant is given by:
[tex] \lambda = \frac{ln(2)}{t_{1/2}} [/tex] (2)
Where:
[tex]t_{1/2}[/tex]: is the half-life of tritium = 12.3 years
By entering equation (2) into (1) we have:
[tex] I_{(t)} = I_{0}e^{-\lambda t} = I_{0}e^{-\frac{ln(2)}{t_{1/2}}t} [/tex]
[tex] \frac{I_{(t)}}{I_{0}} = e^{-\frac{ln(2)}{t_{1/2}}t} [/tex]
By solving the above equation for "t" we have:
[tex] ln(\frac{I_{(t)}}{I_{0}}) = -\frac{ln(2)}{t_{1/2}}t [/tex]
[tex] t = -\frac{ln(\frac{I_{(t)}}{I_{0}})}{\frac{ln(2)}{t_{1/2}}} = -\frac{ln(\frac{1}{10})}{\frac{ln(2)}{12.3}} = 40.9 y [/tex]
Therefore, the watch is 40.9 years old.
I hope it helps you!