How long will it take..

Question

A certain isotope has a half-life of 5.25 days. How long will it take for 30% of the sample to disintegrate? 

 

Solutions are appreciated. Please answer. Arigatou sensei. 

Answer 1

Let A(t) = amount remaining after t days

 

A(t) = A₀e^-kt, where A₀ is the initial amount

 

We know that, in 5.25 days, 1/2 of the original amount remains.

 

So, 0.5A₀ = A₀e^-5.25k

 

        0.5 = e^-5.25k

 

    ln(0.5) = ln(e^-5.25k)

 

     ln(0.5) = -5.25k       k =  0.132

 

A(t) = A₀e^-0.132t

 

Find t so that A(t) = 0.70A₀

 

0.70A₀ = A₀e^-0.132t

 

ln(0.70) = -0.132t

 

t = ln(0.70)/(-0.132) = 2.70 days