Math CentralQuandaries & Queries


Question from aisyah, a student:

if dA/dt = kA
and A(0) = A as the model for the decay of a radioactive substance,
how can show that in general, the half life T of substance is T = -(In 2)/k


Since dA/dt = kA you have dA/A = k dt. A is a function of t so if you integrate both sides you get

ln(A) = kt + C

where C is a constant. Take the exponential of both sides to obtain

A(t) = ekt + C = eC ekt

When t = 0 this gives

A(0) = eC

But you were told that A(0) = A so

A(t) = ekt + C = A ekt

The half life is the time T when A(T) = 1/2 A(0) = 1/2 A. Solve for T.

I hope this helps,

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