In a large section of a statistics class, the points for the final exam are normally distributed, with a mean of 69 and a standard deviation of 9. Grades are assigned such that the top 10% receive A's, the next 20% received B's, the middle 40% receive C's, the next 20% receive D's, and the bottom 10% receive F's. Find the lowest score on the final exam that would qualify a student for an A, a B, a C, and a D.

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.