Exponential decay model!

Question

The half-life of a certain substance is 19 years. How long will it take for a sample of this substance to decay to 60% of its original amount? Use the exponential decay model, A=A₀ e^kt , to solve. 

Answer 1

A(t) = A₀e^kt

• A(t) = amount of substance at time t = 0.6A₀
• A0 = initial amount of substance (at time t=0)
• t = time

 

0.6A₀ = A₀e^kt

0.6 = e^kt

ln(0.6) = kt

ln(0.6)/k = t

 

So what is k?  We could write the half life exponential as:

 

A(t) = A₀(1/2)^t/19

 

So (1/2)^t/19 = e^{ln(1/2)^t/19} = e^{t/19·ln(1/2)} = e^{(ln(1/2)/19)t}

 

So k = ln(1/2)/19 and:

 

t = ln(0.6)/k = 19·ln(0.6)/ln(1/2)

 

Use your calculator to get the answer