.
The formula for decay calculation is:
#A=A_0e^(-lambdaT)#
where #A=# final amount, #A_0=# initial amount, #lambda=# decay constant, and #T=# time.
First, we have to find #lambda#. From the problem statement, we plug in given values and consider #A# to be half of #A_0#:
#1/2A_0=A_0e^(-lambda(24000))#
#1/2=e^(-lambda(24000))#
We take the logarithms of both sides:
#ln(1/2)=lne^(-lambda(24000))#
#ln(1/2)=-lambda(24000)lne=-lambda(24000)#
#lambda=-ln(1/2)/24000=-((-0.69315)/24000)=0.00002888113#
Now, we can use this #lambda# value and solve for time:
#0.2A_0=A_0e^(-0.00002888113T)#
#0.2=e^(-0.00002888113T)#
#ln0.2=lne^(-0.00002888113T)=-0.00002888113Tlne#
#ln0.2=-0.00002888113T#
#T=ln0.2/0.00002888113=(-1.60948)/(-0.00002888113)=55726# years
