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A sample was tested and found to contain 0.32 grams of a radioactive substance with a half-life of four hours. How much of the substance was present in the sample 20 hours before it was tested?

A radioactive substance can undergo radioactive decay, where radioactive decay is the spontaneous emission of radiation by an unstable nucleus. The unstable nucleus emits radiation in an attempt to become more stable. The emitted radiation could be in the form of gamma rays, alpha particles, or beta particles. The radiation can be detected, and the amount of time required for one-half of the radioactive nuclei in the sample to decay can be measured. This is called the half-life.

The sample in the question contains 0.32 grams of a radioactive substance. In the image drawn, the pink dot represents the amount of undecayed radioactive substance, and the orange dots represent the part of the sample that was radioactive but has since undergone radioactive decay. The question tells us that the radioactive substance in the sample has a half-life of four hours. The half-life is the amount of time required for one-half of the radioactive nuclei in a sample to decay. So, after four hours, the number of radioactive nuclei would have halved. But we want to know how much radioactive substance was in the sample before it was tested.

So, if we go back by one half-life or four hours, then the amount of radioactive substance in the sample would double. If we double 0.32 grams, we get 0.64 grams. So, now, we know 0.64 grams of the radioactive substance was present in the sample four hours before it was tested. If we go back by another four hours or another half-life, then the amount of radioactive substance in the sample would double again. If we double 0.64 grams, we get 1.28 grams of radioactive substance. So, we’ve gone back by two half-lives where the half-life is four hours. So, in total, we have gone back eight hours before the sample was tested.

But the question asks us how much of the substance was present in the sample 20 hours before it was tested, so we need to go back further. If we go back another four hours, then the amount of radioactive substance would double again. Double of 1.28 grams is 2.56 grams. So, now, we know that 2.56 grams of the radioactive substance was present in the sample 12 hours before it was tested. If we go back a further four hours or a further half-life, then we find out that 5.12 grams of the radioactive substance was present in the sample 16 hours before it was tested. If we go back a further four hours, then we find that 10.24 grams of radioactive substance is present. We have gone back by four hours, which is the half-life, five times. And since five times four is 20, we have gone back by a total of 20 hours before the sample was tested.

So, the answer to the question how much of the substance was present in the sample 20 hours before it was tested is 10.24 grams.