Uranium ore has two main isotopes, mostly U-238 with just a trace amount of U-235. In a sample of Uranium ore, 99.85% of the atoms are U-238 atoms and 0.15% are U-235 atoms.

Question

Before the Uranium can be used in a nuclear power plant, the proportion of U-235 must be increased to 15% (thus reducing the proportion of U-238 to 85%). This is done by a process called gas diffusion. The ratio of the masses of these two isotopes is U-238 to U-235 = 1.013. Each cycle of the gas diffusion process decreases U-238 by 1.3%. How many cycles are required to reduce the U-238 to 85%?

Answer 1

You want to go from 99.85% U-238 to 85%.

Each cycle of the process decreases the U-238 fraction by a ratio 1-0.013 = 0.987. Let the number of processing cycles required be N.
0.850/0.9985 = 0.8513 = 0.987^N

Solve for N.

N = Log0.8513/Log0.987 = 12.3

Call it 13, for a margin of safety

This is not a calculus problem. Precalc, maybe.

Answer 2

178