exponential decay model

Question

 Use the exponential decay model, A=A o e^kt. The half-life of thorium-229 is 7340 years. How long will it take for a sample of this substance to decay to 20% of its original amount? Round answers to one decimal place.

please upload a picture with steps if possible .

Answer 1

A = A₀e^-kt

 

If the half-life is T, then

 

1/2 = e^-kT

 

ln(0.5) = -kT

 

k = -ln(0.5) / T 

 

A(t) = A₀e^{[ln(0.5) / T]t}

 

A(t) = A₀e^{[ln(0.5) / 7340]t}, t in years

 

If it decayed to 20% of its original amount, there remains 0.20 times the original amount.

 

0.20 = e^{[ln(0.5) / 7340]t}

 

Solve for t.

 

ln(0.20) = [ln(0.5) / 7340]t

 

t = 7340 [ln(0.20) / ln(0.5)] ≅ 17,043.0 years

 

An alternative way to solve this is using

 

A(t) = A₀(1/2)^t/7340, t in years

 

0.20 = (0.5)^t/7340

 

Solve for t.

 

t = 7340 ln(0.20) / ln(0.5) = 17,043.0 years

 

Personally, I think the latter is simpler, but if you need to work in base e (which happens in many physics, chemistry, and engineering problems), then use the given exponential model.