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Chapter 3: Problem 66

The city of Rayburn loses half its population every \(12 \mathrm{yr}\). a) Explain why Rayburn's population will not be zero after 2 half-lives, or 24 yr. b) What percentage of the original population remains after 2 half-lives? c) What percentage of the original population remains after 4 half-lives?

Short Answer

Expert verified
After 2 half-lives, 25% of the original population remains. After 4 half-lives, 6.25% remains.

Step by step solution

01

Understanding Half-Life

The half-life is the time required for a quantity to reduce to half its initial amount. In Rayburn's case, the population halves every 12 years, which means after each half-life, 50% of the current population remains.
02

Calculating Population After 1 Half-Life

After 1 half-life (12 years), the population becomes half of the original population, i.e., 50% remains.
03

Calculating Population After 2 Half-Lives

After 2 half-lives (24 years), the population is halved again. Therefore, 50% of 50% remains, which is \[ 0.5 \times 0.5 = 0.25 \text{ or } 25\% \text{ of the original population.} \]
04

Explaining Why Population is Not Zero

The population decreases exponentially, not linearly, meaning it will continually be halved rather than reaching zero. Each half-life reduces the current population by half, but it never gets to zero theoretically.
05

Calculating Population After 4 Half-Lives

After 4 half-lives, calculate the population using exponential decay: \[ 0.5^4 = 0.0625 \text{ or } 6.25\% \text{ of the original population remains.} \]
06

Conclusion

Rayburn's population can never reach zero theoretically because in each period only half of the current population is reduced. After 2 half-lives, 25% of the original population remains, and after 4 half-lives, 6.25% remains.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Half-Life
The concept of half-life is crucial in illustrating how quantities diminish gradually, particularly in populations or substances. Half-life is the time necessary for an amount to decrease by half its initial quantity. In our case, Rayburn's city population decreases by 50% every 12 years. This is a perfect demonstration of exponential decay, a pattern commonly found in nature and science.
  • The population diminishes by half, not to zero, implying a continuous reduction where each half-life leaves some quantity remaining.
  • After each half-life, the remaining quantity is only half of what it was previously, sustaining a never-ending decrease.
  • This process never results in a complete absence (zero) of the population, highlighting how exponential decay works.
Understanding that half-life results in a halving, not annihilating, any substance or quantity helps in in-depth analyses of diverse scenarios. It's applicable in nuclear physics, demography, and financial decay scenarios, to name a few.
Population Dynamics and Exponential Decay
Population dynamics involve studying how populations change over time, influenced by birth rates, death rates, immigration, and emigration. In Rayburn, we're observing an exponential decay. This describes processes where the rate of change in a population decreases as the population itself lessens.
  • The exponential decay process indicates that the population will continue to decrease, becoming smaller and smaller over time but never truly reaching zero.
  • In 24 years, or two half-lives, the population has diminished to 25% of its original size. This showcases how each half-life episode reduces the population to a fraction of what it was before, halving then halving again.
Population dynamics, when modeled through exponential functions, highlight key perspectives in understanding population declines. Being able to predict these patterns assists in planning resources, studying ecosystems, and understanding long-term demographic trends.
Introduction to Exponential Functions
Exponential functions are mathematical expressions f(x) = a \( b^x \), characterizing constant growth or decay, depending on the base (b). If the base is greater than 1, it's growth; if less than 1, it's decay.
  • These functions are non-linear and indicate rapid changes. Exponential decay, where the decay factor like 0.5 is used, is a typical scenario in reducing populations and substances.
  • In Rayburn's problem, we use the function to model how fractions of the population remain after successive half-lives, indicating significant descent naturally.
  • When considering multiple half-lives, like after 4 half-lives, the population significantly wanes to 6.25% of its original size. This showcases how potent exponential decay can be over multiple timeframes.
Mastering exponential functions is fundamental not just in mathematics, but for real-world applicability in predicting trends, analyzing data, and understanding evolutionary models. This mathematical concept forms the backbone of understanding varying natural and human-induced phenomena.

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