Natural Resource Issues – Water | 自然资源问题——水资源

📚 Natural Resource Issues – Water | 自然资源问题——水资源

Water is essential for life, agriculture, industry and energy production. However, many regions face water scarcity, pollution and unsustainable use. In A Level Mathematics, we can model water resource issues using functions, statistics, calculus and optimisation techniques to understand and predict behaviour, and to inform decision-making.

水对生命、农业、工业和能源生产至关重要。但许多地区面临缺水、污染和不可持续利用等问题。在 A Level 数学中，我们可以利用函数、统计、微积分和优化技术对水资源问题进行建模，从而理解并预测其变化，为决策提供依据。

1. Water Demand and Exponential Growth | 用水需求与指数增长模型

Global water demand has been rising due to population growth and economic development. A simple model assumes that the rate of change of water demand is proportional to the current demand. This leads to the exponential function D(t) = D₀ e^kt, where D₀ is the initial demand, k is the growth constant and t is time in years.

由于人口增长和经济发展，全球用水需求持续上升。一个简单的模型假设用水需求的变化率与当前需求成正比，由此得到指数函数 D(t) = D₀ e^kt，其中 D₀ 为初始需求，k 为增长常数，t 为时间（年）。

If a city’s water use was 1.2 million m³ in 2020 and grows at 3% per year, the demand in 2030 can be predicted as D(10) = 1.2 e^0.03×10 ≈ 1.62 million m³. Conversely, we can solve for k given two data points using logarithms. The doubling time of demand can be found from T_d = ln 2 / k.

如果某城市 2020 年用水量为 120 万 m³，并以每年 3% 的速度增长，2030 年的需求可预测为 D(10) = 1.2 e^0.03×10 ≈ 162 万 m³。反之，给定两个数据点，可利用对数求解 k。需求的翻倍时间可通过 T_d = ln 2 / k 求得。

This exponential model highlights the urgency of water conservation when growth is uncontrolled, as small increases in k lead to dramatically higher demand over time.

该指数模型突显了在增长不受控制时节约用水的紧迫性，因为 k 的微小增加会导致需求量随时间急剧攀升。

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2. Statistical Analysis of Rainfall Data | 降雨数据的统计分析

Annual rainfall in a catchment area can be analysed using measures of central tendency and dispersion. Suppose the mean annual rainfall is 850 mm with a standard deviation of 120 mm. Assuming a normal distribution, we can estimate the probability that rainfall falls below a critical threshold of 700 mm.

流域的年降雨量可以用集中趋势和离散程度的指标进行分析。假设年平均降雨量为 850 毫米，标准差为 120 毫米。若降雨量服从正态分布，我们可以估算降雨量低于临界值 700 毫米的概率。

Using the standard normal variable Z = (X − μ)/σ, we have Z = (700 − 850)/120 = −1.25. From standard normal tables, P(Z < −1.25) ≈ 0.1056, indicating about a 10.6% chance of experiencing less than 700 mm of rain in a given year.

利用标准正态变量 Z = (X − μ)/σ，得到 Z = (700 − 850)/120 = −1.25。查标准正态分布表得 P(Z < −1.25) ≈ 0.1056，即某一年降雨量不足 700 毫米的概率约为 10.6%。

This type of analysis is useful for reservoir planning and drought risk assessment. The interquartile range and box plots can also visualise rainfall variability over decades.

这类分析对于水库规划和干旱风险评估非常有用。四分位距和箱线图也可以直观展示几十年间降雨量的变异性。

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3. Linear Regression for Water Consumption Forecasting | 线性回归预测用水量

Water consumption often correlates with economic indicators such as GDP or industrial output. We can use bivariate data and the least squares regression line y = a + bx to model this relationship. The gradient b represents the change in water use per unit change in the explanatory variable.

用水量通常与 GDP 或工业产值等经济指标相关。我们可以利用双变量数据和最小二乘回归线 y = a + bx 来建模这种关系。斜率 b 表示解释变量每变化一个单位时用水量的变化。

For example, given data on regional GDP (x) and annual water use (y), we compute Σx, Σy, Σx², Σxy and then b = (nΣxy − ΣxΣy) / (nΣx² − (Σx)²) and a = mean(y) − b⋅mean(x). The regression equation can then be used to forecast water demand based on projected economic growth.

例如，给定地区 GDP（x）与年用水量（y）的数据，计算 Σx、Σy、Σx²、Σxy，然后 b = (nΣxy − ΣxΣy) / (nΣx² − (Σx)²)，a = ȳ − b x̄。回归方程便可用于根据预期经济增长预测用水需求。

The strength of the linear relationship is measured by the product moment correlation coefficient r. A value close to 1 indicates a strong positive correlation. Residual analysis helps verify model assumptions and identify outliers that may distort predictions.

线性关系的强度由积矩相关系数 r 衡量。r 值接近 1 表示强正相关。残差分析有助于验证模型假设并识别可能扭曲预测的异常值。

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4. Optimising Water Allocation with Linear Programming | 线性规划优化水资源分配

Regional water authorities must allocate limited water supplies among competing uses such as domestic, agricultural and industrial sectors. Linear programming provides a method to maximise total benefit or minimise cost subject to constraints on water availability, demand and environmental flows.

地区水务部门必须在居民生活、农业和工业等竞争性用水之间分配有限的水资源。线性规划提供了一种在可用水量、需求和环境流量等约束条件下，最大化总效益或最小化成本的方法。

We formulate the problem with decision variables x, y, z for water allocated to each sector. The objective function might be to maximise net revenue: R = c₁x + c₂y + c₃z, subject to constraints such as x + y + z ≤ total supply, x ≥ minimum domestic, etc. Integer programming can also be applied when allocations must be in whole units.

我们用决策变量 x、y、z 表示分配给各部门的水量。目标函数可为最大化净收入：R = c₁x + c₂y + c₃z，约束条件如 x + y + z ≤ 总供水量，x ≥ 最低生活用水等。当分配量必须为整数时，还可应用整数规划。

Graphical methods can be used for two variables, while the simplex algorithm or computational tools handle larger problems. The optimal solution gives the best combination of water allocations under given constraints, and sensitivity analysis reveals how changes in supply affect the ideal allocation.

对于两个变量的情形可用图解法，而单纯形法或计算工具可处理更大规模的问题。最优解给出了在给定约束下水资源分配的最佳组合，敏感性分析则揭示供水量变化如何影响理想分配方案。

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5. Modelling Reservoir Levels with Differential Equations | 用微分方程模拟水库水位

The volume of water in a reservoir changes over time as a result of inflows (rainfall, rivers) and outflows (water supply, evaporation). A simple differential equation for the rate of change of storage S(t) is dS/dt = I(t) − O(t), where I(t) is the inflow rate and O(t) is the outflow rate.

水库中的水量随时间变化，由入流（降雨、河流）和出流（供水、蒸发）引起。描述蓄水量 S(t) 变化率的简单微分方程为 dS/dt = I(t) − O(t)，其中 I(t) 为入流速率，O(t) 为出流速率。

If inflows are constant and outflows depend on the current storage (e.g., proportional to S), we obtain dS/dt = c − kS, which is a first-order linear differential equation. Solving gives S(t) = (c/k) + (S₀ − c/k) e^−kt, showing how storage approaches an equilibrium level over time.

若入流恒定，出流与当前蓄水量成比例（如 O = kS），则得到 dS/dt = c − kS，这是一阶线性微分方程。其解为 S(t) = c/k + (S₀ − c/k) e^−kt，表明蓄水量如何随时间趋近平衡水平。

This model helps predict future storage levels and triggers for water restrictions. More complex models may incorporate seasonal inflow variations using trigonometric functions.

该模型有助于预测未来的蓄水量并为用水限制提供触发条件。更复杂的模型可借助三角函数引入季节性入流变化。

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6. Probability of Drought Events | 干旱事件的概率

Droughts can be modelled as discrete events over time. One approach uses the Poisson distribution if droughts occur randomly and independently at a constant average rate λ per year. The probability of k droughts in a year is P(X = k) = (λ^k e^−λ) / k!.

干旱可建模为随时间发生的离散事件。如果干旱以年均速率 λ 随机且独立发生，可使用泊松分布。一年内发生 k 次干旱的概率为 P(X = k) = (λ^k e^−λ) / k!。

For example, if historical data shows an average of λ = 0.4 droughts per year, the probability of exactly one drought in a year is P(X = 1) = (0.4¹ e^−0.4) / 1! ≈ 0.268. The probability of at least one drought is 1 − P(X = 0) ≈ 1 − 0.670 = 0.

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