a 182.4g sample of germanium-66 is left undisturbed for 22.5 hours. At the end of that period, only 5.70g remain. What is the half-life of this material?

Approximately .

Explanation:

Calculate the ratio between the mass of this sample after and the initial mass:

.

Let denote the number of half-lives in that (where might not necessarily be an integer.) The mass of the sample is supposed to become the previous quantity after each half-life. Therefore, if the initial mass of the sample is (for example,) the mass of the sample after half-lives would be . Regardless of the initial mass, the ratio between the mass of the sample after half-lives and the initial mass should be .

For this question:

.

Take the natural logarithm of both sides of this equation to solve for :

.

.

.

In other words, there are half-lives of this sample in . If the length of each half-life is constant, that length should be .