Problema Solution
The age of the Hawaiian islands gets older as you move from east to west. If the half-life of K-40 is 1.3 billion years, approximate the age of Molokai if a sample rock on the island has 99.90% of its parent K-40 remaining.
Answer provided by our tutors
Let A(0) denote the amount present at t = 0.
Since the half-life is 1.3 billion years we have:
A(1.3) = (1/2)A(0)
Using the formula A(t) = A(0) e^(kt) we have:
A(0)e^(k*1.3) = (1/2)A(0)
e^(1.3k) = 1/2
k = (1/1.3)ln(1/2)
Now we need to find t such that
A(t) = 0.999A(0)
0.999A(0) = A(0) e^(t*((1/1.3)ln(1/2)))
e^(t*((1/1.3)ln(1/2))) = 0.999
(1/2)*e^(t/1.3) = 0.999
e^(t/1.3) = 0.999/0.5
t/1.3 = ln(0.999/0.5)
t = 1.3*ln(0.999/0.5)
t = 0.8998 billion years
1 billion = 1000 million
t = 0.8998 billion years = 899.8 million years
The age of Molokai is 899.8 million years.