A radioactive substance decays at a rate of 30% per year. Presently there are 400 grams of the substance. How long before there are 10 grams?

Question

2279 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-05-28T16:54:42

#ln 40/0.3# years = 12.296 years = 12 years and 108 days, nearly.

If x gm is present at time t years,

# x'=-0.3x# gm/year. Integrating,

#int1/x dx=-0.3int dt#.

So, #ln x=-0.3t + A#. Inversely,

#x=c^(-0.3t+A)= e^Ae^(-0.3t)=Ce^(-0.3t)#

Initially, t = 0 and x = 400. So, C=400, and now,

#x = 400 e^(-0.3t)#

The time t years for decay, from 400 gm to 10 gm, is given by

#10=400e^(-0,3t)#. So, #e^(0.3t)=40#, and inversely,

#0.3t=ln 40#.

So, t=ln 40/0.3=12.296 years= 12 years and 108 days, nearly.