Abreeder reactor converts uranium-238 into an isotope of plutonium-239 at a rate proportional to the amount of uranium-238 present at any time. after 10 years, 0.03% of the radioactivity has dissipated (that is, 0.9997 of the initial amount remains). suppose that initially there is 180 pounds of this substance. find the half-life. (round your answer to the nearest whole number.)

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.

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