Differential equations?

Question

hey, iam having a difficulty in solving this question:

the rate of decay of a radioactive substance is (km) where m is the mass of the substance remaining. Show that the half life (the time in which the amount of the original substance remaining is halved) of the substance is 1/k ln 2

Answer 1

dM/dt = -kM ..... decay rate is kM and M is decreasing with time
dM/M = -kdt ..... put all M terms together
Integrate: ln(M) - ln(M0) = -kt or ln(M/M0) = -kt
We want M = M0/2 so ln(1/2) = -kT where T is the half life
Since -ln(1/2) = ln(2) we get T = (1/k)ln(2)

Answer 2

Let m=m(t) be the mass at time t
with m0 = m(0), initial mass
then dm/dt = -km
solving m = m(0) e^{-kt}
let t = H be half life, so
m(0)/2 = m(0) e^{-kH}
1/2 = e^{-kH}
- kH = ln(1/2) = ln(1) - Ln(2) = -l n(2)
ln(2) = kH
or H = (1/k) ln(2)

Answer 3

. Yes, that's absolutely True! Ferma solved that 'proof' over 45 yrs ago while making the Hydrogen bomb and Super Colider in the White Sands testing grounds.