Mathematical Modelling in Contemporary Urban Environments | 当代城市环境中的数学建模

📚 Mathematical Modelling in Contemporary Urban Environments | 当代城市环境中的数学建模

Contemporary urban environments generate enormous amounts of quantitative data, from population density and air pollution levels to traffic flow and housing prices. In Edexcel A-Level Mathematics, these real-world contexts provide excellent settings for applying pure, statistical and mechanical techniques such as exponential modelling, differentiation, normal distribution and hypothesis testing.

当代城市环境产生了大量的定量数据，从人口密度、空气污染水平到交通流量和房价。在 Edexcel A-Level 数学中，这些现实情境为应用纯数学、统计学和力学技巧提供了绝佳背景，例如指数建模、微分、正态分布和假设检验。

1. Urban Data and Statistical Thinking | 城市数据与统计思维

Urban datasets are rarely neat, but they are ideal for practising statistical thinking. A city council may record hourly traffic counts, daily PM2.5 concentrations, or monthly public transport ridership. Each dataset can be summarised using measures of location and spread: the mean, median, standard deviation and interquartile range.

城市数据集很少是整齐的，但它们非常适合练习统计思维。市政厅可能记录每小时交通流量、每日 PM2.5 浓度或每月公共交通客流量。每个数据集都可以用位置和离散程度的度量来概括：平均值、中位数、标准差和四分位距。

• Mean and median describe the central tendency of urban data.
• Standard deviation and IQR measure how spread out the values are.
• Outliers may indicate special events such as road closures or festivals.

• 平均值和中位数描述城市数据的集中趋势。
• 标准差和四分位距衡量数值的离散程度。
• 异常值可能表明特殊事件，如封路或节日。

When data are approximately symmetric and bell-shaped, the normal distribution becomes a powerful model. In an urban environment, variables such as the daily number of metro passengers often follow an approximately normal pattern after seasonal adjustment.

当数据近似对称且呈钟形时，正态分布就成为一种强大的模型。在城市环境中，经过季节调整后的每日地铁乘客数等变量往往近似服从正态分布。

2. Modelling Population Growth | 人口增长建模

Urban populations can grow rapidly. If a city has population P₀ at time t = 0 and grows at a constant relative rate k per year, the population after t years is modelled by the exponential function:

城市人口可以快速增长。如果一座城市在 t = 0 时人口为 P₀，并且每年以恒定的相对增长率 k 增长，那么 t 年后的人口可以用指数函数建模：

P(t) = P₀eᵏᵗ

For example, if P₀ = 500 000 and k = 0.02, then P(10) = 500 000 × e⁰·² ≈ 610 701. This model assumes unlimited resources, so it may overestimate growth in a constrained urban area.

例如，如果 P₀ = 500 000 且 k = 0.02，那么 P(10) = 500 000 × e⁰·² ≈ 610 701。该模型假设资源无限，因此可能高估了受限城市区域的人口增长。

To find the growth constant k from data, use logarithms. Taking natural logs of both sides gives ln P = ln P₀ + kt, which is a straight line with gradient k when ln P is plotted against t.

要根据数据求增长率常数 k，可以使用对数。对两边

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