The two lines represent the amount of water, over time in two tanks that are the same size. Which container is filling more slow
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Let denote the amount of uranium-238 in the reactor at time
. As conversion to plutonium-239 occurs, the amount of uranium will decrease, so the conversion rate is negative. Because the rate is proportional to the current amount of uranium, we have
where is constant. Separating variables and integrating both sides gives
Suppose we start some amount . This means that at time
we have
, so that
We're given that after 10 years, 99.97% of the original amount of uranium remains. This means (if is taken to be in years) for some starting amount
,
The half-life is the time it takes for the starting amount
to decay to half,
:
or about 23,101 years. Notice that it doesn't matter what the actual starting amount is, the half-life is independent of that.