PRE CALC AB - A decaying radioactive ore has a half life of 1710 years.

Question

A decaying radioactive ore has a half life of 1710 years.

a) Write the exponential decay function for a 27 gram sample.

b) Find the amount remaining in 3000 years.

Answer 1

By noting the half-life as λ, we know the following representation which relates the amount remaining after an amiunt of time to the original amount as...

A(t) = A₀e^(-λt)

A(t)/A₀ represents the percentage of the sample remaining. Lets call this ratio R.

So, ln(A(t)/A₀) = ln(e^(-λt)) --> ln(R) = -λt. Since 0 <= R <= 1, the ln(R) = -ln(1/R)

So ln(1/R) / t = λ

But we know in 1710 years, we have 50% of the sample remaining (1/2 = R).

Therefore --> ln(2)/1710 = λ

This is the decay rate.

The equation would be ...

A = 27e^(- [ln(2)/1710] * t)

Now plugging in 3000 years for t will tell us the amount of the sample remaining after that amount of time.

A₃₀₀₀ = 27 e^(- [ln(2)/1710] *3000) ≈ 8.00 grams