Use Logarithms to solve

Question

A certain element has a half life of
4.5
billion years.

a. You find a rock containing a mixture of the element and lead. You determine that
20%
of the original element remains; the other
80%
decayed into lead. How old is the rock?

b. Analysis of another rock shows that it contains 60% of its original element; the other 40% decayed into lead. How old is the rock?

Answer 1

A(t) = amount present at time t, with t in billions of years

T = half life in billions of years = 4.5

A₀ = initial amount (at t = 0)

 

A(t) = A₀ (0.5)^t/T

 

At age t, A(t) = 0.20A₀

 

0.20A₀ = A₀ (0.5)^t/4.5

 

0.2 = (0.5)^t/4.5

 

log(0.2) = (t/4.5)log(0.5)

 

t = 4.5 log(0.2)/log(0.5) ≅ 10.45

 

The age of the rock is about 10.45 billion years.