Half-life of approx. 3.45 years.

Question

Show that a radioactive substance which exponentially decays to 20% of its initial value after 8 years has a half-life of approximately 3.45 years. Recall that the half-life is the length in time it takes an exponentially-decaying function to be reduced to half of its original quantity.

Answer 1

decays to 20% in 8 years

A = Pe^rt is the general formula

.2 = e^8r

ln.2 = 8r

r = ln.2/8 = about -.2012 = about negative 20%

half life is when

.5 = e^.2t

ln.5 = .2t

t= ln.5/.2 = about 3.467

or use r=-.2012

then

t = ln.5/.2012 = about 3.445=about 3.45 years for the half life