📚 The World’s Population | 世界人口
Understanding how the world's population grows is not just a topic for geography or economics – it is a powerful application of exponential functions and logarithms in IGCSE Mathematics. In the Edexcel syllabus, you will often encounter real-life problems about population growth, doubling times, and predictions. This article explores the mathematical models behind the world’s population, linking historical data with the exponential growth model, and shows you how to use logarithms to solve typical exam questions.
理解世界人口如何增长不仅仅属于地理或经济学的范畴——它在 IGCSE 数学中是指数函数与对数的一个强有力的应用。在 Edexcel 考试大纲中，你经常会遇到关于人口增长、倍增时间以及预测的现实情境问题。本文将探讨世界人口背后的数学模型，将历史数据与指数增长模型联系起来，并展示如何运用对数去解答典型的考题。
1. Introduction to World Population Growth | 世界人口增长简介
The global population has grown dramatically over the past two centuries. From about 1 billion in 1800, it reached 8 billion in November 2022. This type of increase, where the growth rate is roughly proportional to the current population, suggests an exponential pattern. In mathematics, we model such behaviour using exponential functions, a key topic in the IGCSE Edexcel curriculum.
全球人口在过去两个世纪中急剧增长。从 1800 年大约 10 亿人，到 2022 年 11 月突破 80 亿。这种增长大致与现有人口成正比的模式，显示出一种指数规律。在数学中，我们使用指数函数对这种行为进行建模，这也是 IGCSE Edexcel 课程中的一个核心主题。
2. Historical Data and Trends | 历史数据与趋势
Looking at some key milestones helps us appreciate the acceleration of population growth. The table below shows the years when the world’s population reached new billions.
浏览一些关键的里程碑有助于我们感受人口增长的加速。下表展示了世界人口达到新增 10 亿人的年份。
| Year | Population (billions) |
|---|---|
| 1800 | 1 |
| 1927 | 2 |
| 1960 | 3 |
| 1974 | 4 |
| 1987 | 5 |
| 1999 | 6 |
| 2011 | 7 |
| 2022 | 8 |
Notice how the time taken to add each extra billion has shortened for much of the 20th century. This behaviour can be approximated by a continuous exponential curve, making it ideal for IGCSE modelling questions.
请注意，在 20 世纪的大部分时间里，每增加 10 亿人口所需的时间都在缩短。这种行为可以用一条连续的指数曲线来近似描述，非常适合 IGCSE 的建模问题。
3. The Exponential Growth Model | 指数增长模型
Exponential growth occurs when the rate of change of a quantity is proportional to the quantity itself. For the world’s population, we can write the model in two common forms used in Edexcel IGCSE:
当一个量的变化率与其本身成正比时，就会发生指数增长。对于世界人口，我们可以用 Edexcel IGCSE 中常见的两种形式写出模型：
P(t) = P₀ × (1 + r)t
where P₀ is the initial population, r is the annual growth rate (as a decimal), and t is the time in years. If growth is continuous, the model becomes:
其中 P₀ 为初始人口，r 为年增长率（用小数表示），t 为经过的年数。如果增长是连续的，模型变为：
P(t) = P₀ ekt
In this version, k is the continuous growth rate. Both forms are regularly examined, and you must be comfortable using the one stated in the question.
在这个版本中，k 是连续增长率。两种形式都会经常考查，你必须熟练使用题目中给出的那种形式。
4. Growth Rate and Doubling Time | 增长率与倍增时间
determines how fast the population multiplies. A very useful concept is the doubling time – s for the population to double. For the discrete model, we set P(t) = 2P₀ and solve for t:
年增长率 r 或连续增长率 k 决定了人口翻倍的速度。一个非常有用的概念是倍增时间，即人口翻一番所需的年数。对离散模型，我们设 P(t) = 2P₀ 并求解 t：
2P₀ = P₀ (1 + r)T → T = log 2 / log(1 + r)
For continuous compounding, the doubling time is expressed simply as:
对于连续复合增长的模型，倍增时间可简明地表示为：
Td = ln 2 / k
A rough estimate often used is the 'rule of 70': doubling time ≈ 70 ÷ (percentage growth rate). For example, at a growth rate of 1.1% ately 70/1.1 ≈ 63.6 years. The exact calculation using the formula gives about 63.0 years.
常用的粗略估算是“70 法则”：倍增时间 ≈ 70 ÷（增长率百分数）。例如，年增长率为 1.1%，倍增时间约为 70/1.1 ≈ 63.6 年。而用公式精确计算得出的结果约为 63.0 年。
5. Using Logarithms in Population Calculations | 对数在人口计算中的应用
ponential equation. This is where logarithms come in. Given a model like P(t) = P₀ (1 + r)t, and target population P, you can isolate t:
当 IGCSE 题目要求你预测人口达到某个特定值的年份时，你就需要求解指数方程。这正是对数的用武之地。给定模型 P(t) = P₀ (1 + r)t 以及目标人口 P，你可以将 t 分离出来：
t = log(P / P₀) / log(1 + r)
You may use either common logarithms (base 10) or natural logarithms (ln), as long as you are consistent. Edexcel allows both, and your calculator will handle either.
你可以使用常用对数（以 10 为底）或自然对数（ln），只要保持前后一致即可。Edexcel 允许两种方法，你的计算器都能处理。
6. Predicting Future Population | 预测未来人口
Let' Suppose the world’s population was 8.0 billion in 2022, with an annual growth rate of 0.84%. We can estimate the population in 2050. Firstly, t = 2050 – 2022 = 28 years, and r = 0.0084. Using the compound growth formula:
让我们用模型做一个真实世界的预测。假设 2022 年世界人口为 80 亿，年增长率为 0.84%。我们可以估算 2050 年的人口。首先，t = 2050 – 2022 = 28 年，r = 0.0084。使用复合增长公式：
P(28) = 8.0 × (1.0084)28
Calculating (1.0084)28 lion. So the model projects the population will exceed 10 billion by 2050. Working backwards, you can also find when we would hit 10 billion.
计算 (1.0084)28 ≈ 1.264，得到 P ≈ 8.0 × 1.264 = 101 亿。因此该模型预测到 2050 年人口将超过 100 亿。反推时，你也可以求出达到 100 亿的时间点。
7. Limitations of the Model | 模型的局限性
Exponential models assume unlimited resources and constant growth rates, which cannot hold indefinitely. In reality, factors like food supply, disease, wars, and changing birth rates slow down growth. A more realistic population model is the logistic growth model, but that is beyond the scope of IGCSE. For exam purposes, you are only expected to use the exponential models given in the question, while being aware that they are approximations.
指数模型假设资源无限且增长率恒定，这在现实中不可能永远成立。实际上，食物供应、疾病、战争以及生育率变化等因素都会减缓增长。更现实的人口模型是逻辑斯蒂增长模型，但这已超出 IGCSE 的范围。在考试中，你只需使用题目给出的指数模型，同时意识到它们是近似的结果。
8. Worked Example and Application | 例题与应用
Question: In 2022, the population of the world was 8.0 billion. The annual growth rate was 0.9%. Find the year when the population is expected to reach 10 billion, assuming the rate remains unchanged.
题目：2022 年世界人口为 80 亿，年增长率为 0.9%。假设增长率保持不变，求人口预计达到 100 亿的年份。
Solution: Start with the formula P = P₀(1 + r)t. Here P₀ = 8.0, r = 0.009, and P = 10.0. Substitute:
解答：从公式 P = P₀(1 + r)t 出发。这里 P₀ = 8.0，r = 0.009，P = 10.0。代入：
10.0 = 8.0 × (1.009)t
Divide both sides by 8.0: (1.009)t = 1.25. Now take logarithms:
两边除以 8.0：(1.009)t = 1.25。现在取对数：
t = log(1.25) / log(1.009)
ves the year 2022 + 24.9 ≈ 2047. Thus, the population should reach 10 billion around 2047.
使用计算器，log(1.25) ≈ 0.09691，log(1.009) ≈ 0.003891，所以 t ≈ 0.09691 / 0.003891 ≈ 24.9 年。加上 2022 得到 2022 + 24.9 ≈ 2047。因此，人口应在 2047 年前后达到 100 亿。
9. Real-World Population Challenges | 现实世界的人口挑战
Beyond the mathematics, rapid population growth creates challenges for food production, water resources, and the environment. As a student, recognising the connection between the exponential model and these real issues makes the topic more meaningful. It also helps you interpret answers in context – for example, a prediction of 100 billion people by 2100 would be unrealistic, indicating an incorrect model or assumption.
在数学之外，人口的快速增长给粮食生产、水资源和环境带来了挑战。作为学生，认识到指数模型与现实问题之间的联系会让这一课题更有意义。这还有助于你在情境中解释答案——例如，预测到 2100 年人口达到 1000 亿是不切实际的，这就表明模型或假设有误。
10. Summary and Exam Tips | 小结与备考提示
When facing a population question in the Edexcel IGCSE Mathematics exam, remember these key points:
在 Edexcel IGCSE 数学考试中遇到人口问题时，请牢记以下要点：
Identify the model: Is it P = P₀(1+r)t or P = P₀ekt? Read carefully.
识别模型：是 P = P₀(1+r)t 还是 P = P₀ekt？仔细审题。
Convert percentages: A rate of 1.2% must be written as r = 0.012.
转化百分比：1.2% 的增长率必须写成 r = 0.012。
Use logarithms correctly: Take logs of both sides, and be consistent with your chosen base.
正确使用对数：两边取对数，并保持所用底数一致。
Check your answer: Does the result make sense in the real world? If t is negative or ridiculously large, reconsider your working.
检验答案：结果在现实世界中合理吗？如果 t 为负数或大得不合理，请重新检查计算过程。
Doubling time: Useful for quick estimates; formulae are given but you must know how to apply them.
倍增时间：用于快速估算；公式会给出，但你必须知道如何应用。
With practice, population growth problems become an excellent opportunity to secure marks on exponential and logarithmic questions.
通过练习，人口增长问题将成为你在指数与对数题目中稳拿分数的绝佳机会。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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