Answer: 9595.8 years
Explanation:
Half life is the amount of time taken by a radioactive material to decay to half of its original value.
[tex]t_{\frac{1}{2}}=\frac{0.693}{k}[/tex]
[tex]k=\frac{0.693}{5750}=1.2\times 10^{-4}years^{-1}[/tex]
Expression for rate law for first order kinetics is given by:
[tex]t=\frac{2.303}{k}\log\frac{a}{a-x}[/tex]
where,
k = rate constant
t = age of sample
a = let initial amount of the reactant =100
x = amount decomposed or lost = [tex]\frac{69.4}{100}\times 100=69.4[/tex]
a - x = amount left after decay process= (100-69.4)= 30.6
Putting in the values:
[tex]t=\frac{2.303}{1.2\times 10^{-4}}\log\frac{100}{30.6}[/tex]
[tex]t=9595.8years[/tex]
Thus the bones were 9595.8 years old.