17 歐拉數e的魔力 umber e or a whisper fading away. It also simplifies complex problems, making it a go-to tool for scientists and engineers. In biology, e models population growth. Picture a colony of bacteria doubling every hour. The smooth, accelerating curve of their growth uses e to predict how many bacteria there will be after a day or a week [4]. In ecology, e helps track how animal populations expand or how resources, like fish stocks, shrink when overharvested. For example, conservationists use e to estimate how quickly a threatened species might recover if protected [4]. In physics and chemistry, e describes decay, such as how radioactive elements like uranium lose energy over time. Scientists rely on e to calculate a substance’s half-life, the time it takes for half of it to break down, which is crucial for safe handling in medical treatments or power plants. In daily life, e shines in engineering and technology that makes our life easier. It models how a capacitor stores charge in a circuit, essential for designing devices like your phone or computer. In medicine, e helps track how drugs are absorbed or cleared from the body, helping doctors determine safe dosages. In technology, e underpins algorithms for signal processing, ensuring clear audio in your earbuds or smooth video streaming. Enormous Impact of the Seemingly Small Number What makes e truly special is that it connects these diverse phenomena. Its value, about 2.718, may seem small, but its impact is enormous. Next time you hear about a virus spreading, a species recovering, or a cup of tea cooling, think of e — a quiet number with a massive role in unlocking the secrets of our world. A Practical Archeological Question: Carbon-14 Dating Carbon-14 dating is a method used to determine the age of organic archeological specimens from the age of 500 to 50,000 years [5]. Carbon-14 is an unstable radioisotope, which undergoes decay into nitrogen-14. Living plants incorporate naturally occurring atmospheric carbon-14 into their tissues through carbon fixation, and pass it on to animals through the food chain. The ratio of carbon-14 in living tissues is relatively stable because living organisms constantly take in air and food despite the constant decay of carbon-14, but once the organism dies, there will be a net reduction in carbon-14 content. The decay process can be expressed by the exponential decay function: N = N0e-kt, where N is the number of undecayed nuclei, N0 is the initial number of undecayed nuclei, k is the decay constant, and t is the time lapsed [6, 7]. The half-life t1/2 of carbon-14, or the time needed for half of the radioisotope to decay, is roughly 5,730 years. If we have a piece of ancient wood whose carbon-14 content has four tenths of that in living trees, find: 1) the decay constant k (to four decimal places). 2) the age of that piece of wood (to the nearest year). Solution:
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