SOLUTION: It is known that the mass, m(t), of a radio-active substance decreases as it decays, and that the equation governing this is m'(t)=-.005m(t) where t is in years. What is the half-l

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Question 256914: It is known that the mass, m(t), of a radio-active substance decreases as it decays, and that the equation governing this is m'(t)=-.005m(t) where t is in years. What is the half-life?
any help would be great!!
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Half life means half of the original amount remains.
dm%2Fdt=-0.005m
dm%2Fm=-0.005%2Adt
Integrate,
ln%28m%29=-0.005t%2BC
m=Ce%5E%28-0.005t%29
where C=m0 when t=0, the original amount of material.
m=m0%2Ae%5E%28-0.005t%29
At t=0, m=m0
When t=t_h, m=%281%2F2%29m0
Substitute,
%281%2F2%29m0=m0%2Ae%5E%28-0.005%28t_h%29%29
1%2F2=e%5E%28-0.005%28t_h%29%29
ln%281%2F2%29=-0.005%28t_h%29
t_h=%28ln%281%2F2%29%2F%28-%280.005%29%29%29
t_h=138.6 years, approximately