Suppose you have a rock that, when it solidifies, contains 1 microgram of a radioactive isotope. how much of this isotope remains after five half-lives?

The original mass is m₀ = 1 μg.

The decay equation is

where
m = mass remaining after time t,
k = constant.

Let
t = t₁ = time for half-life.
Then


When t = 5 half-lives, then
t = 5*(0.6931/k) = 3.4657/k
The mass remaining is


The mass remaining after 5 half-lives is
0.0313 μg

0.0313 μg

They can measure  the amount of energy gained or lost