Answer:
t = 8588 years
Step-by-step explanation:
From the given information:
Using the formula:
[tex]k = \dfrac{In (2)}{t_{1/2}}[/tex]
Given that:
[tex]t_{1/2}[/tex] = 5730 years
Then:
[tex]k = \dfrac{In (2)}{5730}[/tex]
[tex]k = \dfrac{0.6931472}{5730}[/tex]
k = 1.20968 × 10⁻⁴
However; the expression to find the value of x is:
[tex]x= x_oe^{-kt}[/tex]
[tex]4.6 \times 10^{-13} = 1.3 \times 10^{-12} \times e^{(-1.20968\times 10^{-4} \times t)}[/tex]
[tex]\dfrac{4.6 \times 10^{-13} }{1.3 \times 10^{-12}}= e^{(-1.20968\times 10^{-4} \times t)}[/tex]
[tex]0.353846= e^{(-1.20968\times 10^{-4} \times t)}[/tex]
(-1.20968 × 10⁻⁴ × t) = In(0.353846)
(-1.20968 × 10⁻⁴ × t) = -1.03889
[tex]t = \dfrac{-1.03889}{-1.20968 \times 10^{-4}}[/tex]
t = 8588 years
