Mathematics of Sustainable Urban Development | 可持续城市发展的数学

📚 Mathematics of Sustainable Urban Development | 可持续城市发展的数学

Sustainable urban development requires balancing economic growth, social equity and environmental protection. A-level Mathematics provides essential tools for quantifying these challenges, from population growth and resource consumption to air quality and traffic flow. This article explores how core Edexcel A-level Maths skills – calculus, statistics, algebra and optimisation – can be used to model and evaluate sustainable urban policies.

可持续城市发展需要在经济增长、社会公平和环境保护之间取得平衡。A-level 数学为量化这些挑战提供了重要工具，从人口增长和资源消耗到空气质量和交通流量。本文探讨如何运用 Edexcel A-level 数学的核心技能——微积分、统计、代数和优化——来建模和评估可持续城市政策。

1. Modelling Urban Population Growth | 城市人口增长建模

A simple starting point is the exponential growth model. If a city has initial population P₀ and grows at a continuous rate k per year, the population after t years is given by P(t) = P₀ e^(kt). This arises from the differential equation dP/dt = kP, meaning the growth rate is proportional to the current population. In A-level Maths, you learn to solve such equations by separation of variables.

一个简单的起点是指数增长模型。如果一个城市的初始人口为 P₀，每年以连续速率 k 增长，那么 t 年后的人口由 P(t) = P₀ e^(kt) 给出。这源自微分方程 dP/dt = kP，即增长率与当前人口成正比。在 A-level 数学中，你将学习通过分离变量法求解这类方程。

P(t) = P₀ e^(kt)

However, unlimited exponential growth is not sustainable. The doubling time is found by setting P(t) = 2P₀, giving t = (ln 2)/k. This is a useful exam-style calculation, but real cities face resource limits.

然而，无限的指数增长是不可持续的。翻倍时间通过令 P(t) = 2P₀ 求得，得到 t = (ln 2)/k。这是一个实用的考试型计算，但真实城市面临资源限制。

2. Logistic Growth and Carrying Capacity | 逻辑斯蒂增长与承载能力

The logistic model introduces a carrying capacity K, the maximum population an urban area can support sustainably. The differential equation is dP/dt = rP(1 − P/K). The solution is P(t) = K / (1 + ((K − P₀)/P₀) e^(−rt)). As t increases, P(t) approaches K, so growth slows near the limit.

逻辑斯蒂模型引入承载能力 K，即一个城市地区能够可持续支持的最大人口。微分方程为 dP/dt = rP(1 − P/K)。其解为 P(t) = K / (1 + ((K − P₀)/P₀) e^(−rt))。随着 t 增大，P(t) 趋近于 K，因此增长在接近极限时放缓。

dP/dt = rP(1 − P/K)

In exams, you may be asked to find the limiting population by letting t → ∞ or to show that the growth rate is maximised at P = K/2. This links differentiation to urban sustainability.

在考试中，你可能会被要求通过让 t → ∞ 求极限人口，或证明增长率在 P = K/2 时最大。这将微分与城市可持续性联系起来。

3. Optimising Urban Energy Use with Linear Programming | 用线性规划优化城市能源使用

Linear programming (LP) is a decision-making tool that appears in Edexcel Decision Mathematics and can be adapted to core algebra. Consider a city choosing between two renewable energy sources: solar (x MWh) and wind (y MWh). Constraints might include land area, budget and minimum output. The objective could be to minimise carbon emissions C = ax + by subject to constraints like 2x + 3y ≤ 120 and x + y ≤ 50, x ≥ 0, y ≥ 0.

线性规划 (LP) 是一种决策工具，出现在 Edexcel 决策数学中，也可适用于核心代数。设想一个城市在两种可再生能源之间选择：太阳能（x MWh）和风能（y MWh）。约束条件可能包括土地面积、预算和最低产出。目标可能是最小化碳排放 C = ax + by，约束条件如 2x + 3y ≤ 120 和 x + y ≤ 50，x ≥ 0，y ≥ 0。

Plotting the feasible region and testing vertices gives the optimal solution. This type of question tests your ability to interpret inequalities graphically and to evaluate objective functions.

绘制可行区域并检验顶点可以得到最优解。这类题目考查你用图形解释不等式以及计算目标函数的能力。

4. Statistical Analysis of Air Quality Data | 空气质量数据的统计分析

Urban air quality is often measured by PM2.5 concentration. Suppose a city’s daily PM2.5 readings (μg m⁻³) are normally distributed with mean μ and standard deviation σ. A-level Statistics requires calculating probabilities such as P(X > 35) using the z-score z = (x − μ)/σ.

城市空气质量通常以 PM2.5 浓度来衡量。假设一个城市的每日 PM2.5 读数（μg m⁻³）服从均值为 μ、标准差为 σ 的正态分布。A-level 统计学要求使用 z 分数 z = (x − μ)/σ 计算概率，例如 P(X > 35)。

z = (x − μ) / σ

You may also be asked to estimate μ and σ from a sample using the formulae x̄ = Σx/n and s = √(Σ(x − x̄)²/(n−1)). These statistics help policymakers decide whether interventions are working.

你或许还需要使用公式 x̄ = Σx/n 和 s = √(Σ(x − x̄)²/(n−1)) 从样本中估计 μ 和 σ。这些统计量帮助政策制定者判断干预措施是否有效。

5. Probability Distributions in Traffic Flow | 交通流量中的概率分布

Traffic arrivals at a junction can be modelled by a Poisson distribution. If the mean number of vehicles per minute is λ, then P(X = k) = e^(−λ) λ^k / k!. For example, if λ = 4, the probability of exactly 6 arrivals in a minute is calculated by substituting k = 6.

路口到达的车辆可以用泊松分布建模。如果每分钟平均车辆数为 λ，

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