Mathematics

Problem

Let

Q(t)=Q_0e^{kt}

be the general solution of the differential equation

\dfrac{dQ}{dt}=kQ

, where

k

is a nonzero constant. (a) Derive an explicit formula for the half-life

T_{1/2}

(the time required for

Q

to be reduced to

\tfrac{1}{2}Q_0

) in terms of

k

. (b) Explain why

k

must be negative in any physical decay process and interpret the magnitude of

k

with respect to the rate at which the substance decays.