# How Are Radioactive Decay Problems Solved?

Radioactive decay problems are solved by using a formula for exponential decay where the final amount of radioactive material equals the initial amount times e to the power of k times time. Simple substitution of the known values will yield the unknown value.

The formula for exponential decay is written as A=A0e^kt, where A0 represents the initial amount of radioactive material, A is the final amount of material, k is a constant indicative of half-life, t is the time and the ^ symbol means to the power of. The symbol e is a mathematical concept that stands for base of a natural logarithm.

A typical radioactive decay problem might say, after two days, a sample of carbon-14 has decayed 75%, so what is the half-life? To make the problem easy, assume 100 grams for the original mass. So 75 = 100e^2k, or 0.75=e^2k.

Take the ln (the log) of both sides. A graphing calculator gives the ln of the left side of the equation. The ln of e is equal to the power of e. So the ln of e^2k equals 2k. Combined, the equation becomes -0.3=2k, with k equal to -0.15.

To get the half life, plug k into the formula when A=1/2A0 and solve for t. In this case, the half-life is 4.67 days. The formula can also be simplified to A=A0*2^(-t/h), where h is the half-life. However, the first method is more accurate.

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