The answer is 6×10⁹ years.
K-Ar dating is radioactive dating method used to measure a time elapsed since the rock is cool enough to trap Ar after a radioactive decay of 40K to 40Ar. It can be expressed as:
\(t= \frac{ t_{ \frac{1}{2} } }{ln(2)} ln( \frac{ K_{f} + \frac{ Ar_{f} }{0.109} }{ K_{f} } )\)
where:
\(t\) - elapsed time
\(t_{ \frac{1}{2} }\) - half-life of K40
\(K_{f}\) - amount of K40
\(Ar_{f}\) - amount of Ar40
We know that a half-life of K40 is 1.251×10⁹ years:
\(t_{ \frac{1}{2} } =1.251* 10^{9}\)
We do not know absolute value of amount of K40 and Ar40, but we know percentage and can express them as following:
\(K_{f} =25%=0.25\)
\(Ar_{f} =75%=0.75\)
So:
\(t= \frac{1.251* 10^{9} }{ln(2)} ln( \frac{0.25+ \frac{0.75}{0.109} }{0.25} )\)
⇒ t = 6×10⁹
Thus, the rock is old 6×10⁹ years.
