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In his article Light Attenuation and Exponential Laws in the last issue of Plus, Ian Garbett discussed the phenomenon of light attenuation, one of the many physical phenomena in which the exponential function crops up. In this second article he describes the phenomenon of radioactive decay, which also obeys an exponential law, and explains carbon dating exponential decay this information allows us to carbon-date artefacts such as the Dead Sea Scrolls. In the previous article, we saw that light attenuation obeys an exponential law. To show this, we needed to make one critical assumption: that for a thin enough slice of matter, the proportion of light getting through the slice was proportional to the thickness of the slice.
Carbon dating exponential decay
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In this section we will explore the use of carbon dating to determine the age of fossil remains. Carbon is a key element in biologically important molecules. During the lifetime of an organism, carbon is brought into the cell from the environment in the form of either carbon dioxide or carbon-based food molecules such as glucose; then used to build biologically important molecules such as sugars, proteins, fats, and nucleic acids. These molecules are subsequently incorporated into the cells and tissues that make up living things. Therefore, organisms from a single-celled bacteria to the largest of the dinosaurs leave behind carbon-based remains. Carbon dating is based upon the decay of 14 C, a radioactive isotope of carbon with a relatively long half-life years. While 12 C is the most abundant carbon isotope, there is a close to constant ratio of 12 C to 14 C in the environment, and hence in the molecules, cells, and tissues of living organisms. This constant ratio is maintained until the death of an organism, carbon dating exponential decay 14 C stops being replenished. At this point, the overall amount of 14 C in the organism begins to decay exponentially. Therefore, by knowing the amount of 14 C in fossil remains, you can determine how long ago an organism died by examining the departure of the observed 12 C to 14 C ratio from the expected ratio for a living organism.
In real-world applications, we need to model the behavior of a function. In mathematical modeling, we choose a familiar general function with properties that suggest that it will model the real-world phenomenon we wish to analyze. In the case of rapid growth, we may choose the exponential growth function:. We may use the exponential growth function in applications involving doubling timethe time it takes for a quantity to double. Such phenomena as wildlife populations, financial investments, biological samples, and natural resources may exhibit growth based on a doubling time. In some applications, however, as we will see when we discuss the logistic equation, the logistic model sometimes fits the data better than the exponential model. On the other hand, if a quantity is falling rapidly toward zero, without carbon dating exponential decay reaching zero, then we should probably choose the exponential decay model.
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