Using the formula:
#Q(t) = Q(0)e^(alphat)#
Using this reference half-life of #""^42K#, set #t = 12.36" hours"# and make #(Q(t))/(Q(0)) = 1/2#:
#1/2 = e^(alpha(12.36" hours"))#
Use the natural logarithm on both sides:
#ln(1/2) = ln(e^(alpha(12.36" hours")))#
Flip the equation and ln(e) redues to the exponent:
#alpha(12.36" hours") = ln(1/2)#
#alpha(12.36" hours") = -ln(2)#
#alpha = -ln(2)/(12.36" hours"#
#Q(62" hours") = (500 "g of "^42K)e^((-ln(2)/(12.36" hours")(62" hours"))#
#Q(62" hours") = 15.45 "g of "^42K#
