radioactive isotope leaked into a stream

Question

A certain radioactive isotope has leaked into a small stream. Three hundred days after the​ leak,

55​% of the original amount of the substance remained. Determine the​half-life of this radioactive isotope.

Answer 1

if the governing equation is of the form S = S_oe-kt

where

S is the amount of material over time

S_o = the amount of material at time = 0

k is the decay constant

t = time (days)

Therefore, lets assume we have 100 lbs of material (S_o) to start off with (this is strictly arbitrary!)

Then, 55 = 100e^(-300t) or .55 = e^(-300k) then ln(.55) = -300k

and k = .002

To find the half life, we then find the time when 1/2 remains, like so

50 = 100e^(-.002t) then .5 = e^(-.002t) Solving for t, we find that

t = 346.6 days = the half life!