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Shallow Green: Optimising a Green Roof with Edexcel A-Level Calculus | 浅绿屋顶：用 Edexcel A-Level 微积分优化绿色屋顶

2026-08-29 21:27 · 本文部分内容由AI辅助创作

📚 Shallow Green: Optimising a Green Roof with Edexcel A-Level Calculus | 浅绿屋顶：用 Edexcel A-Level 微积分优化绿色屋顶

A shallow green roof is not just an environmental feature; it is also a powerful context for A-Level mathematical modelling. By combining integration, differentiation, exponential decay and optimisation, we can design a roof with a thin growing layer that meets practical constraints such as maximum depth, volume and cost.

浅绿屋顶不仅是一种环保设施，也是 A-Level 数学建模的有力情境。通过综合运用积分、微分、指数衰减和优化，我们可以设计出满足最大深度、体积和成本等实际约束的薄层绿色屋顶。

1. Understanding the Problem | 理解问题

green roof has a thin layer of substrate or soil placed on a waterproof membrane. The term 'shallow green' If the substrate depth is constant at d metres, the volume of substrate is simply V = LWd.

浅绿屋顶在防水膜上铺设一层薄薄的基质或土壤。“浅绿”指的是种植介质深度较小。设屋顶长度为 L 米，宽度为 W 米。如果基质深度恒定为 d 米，则基质体积为 V = LWd。

In many exam questions the depth is not constant, and the cross-section is described by a function d(x). The task is then to use integration to find the volume and differentiation to optimise the design.

在许多考试题中，深度并不是恒定的，横截面由一个函数 d(x) 描述。此时任务就是用积分求体积，用微分优化设计。

2. Modelling the Roof Profile | 屋顶剖面建模

d(x) = ax² + bx + c, where d(x) is the depth in metres at position x metres from one edge.

设屋顶宽度在 x 轴上对应区间 0 ≤ x ≤ W。基质深度沿宽度变化。我们可以用二次函数 d(x) = ax² + bx + c 来模拟横截面，其中 d(x) 是距离一侧边缘 x 米处的深度（单位：米）。

d(x) = ax² + bx + c, 0 ≤ x ≤ W

The coefficients a, b and c are chosen to satisfy given conditions, such as zero depth at the edges or a maximum depth at the centre.

系数 a、b 和 c 根据给定条件确定，例如边缘深度为零或中心深度最大。

3. Area of the Shallow Green Layer | 浅绿层面积

x) is positive on the interval [0, W], the area A is given by the definite integral A = ∫₀W d(x) dx.

种植介质的横截面积通过对深度函数在宽度上积分得到。如果 d(x) 在区间 [0, W] 上为正，则面积 A 由定积分 A = ∫₀W d(x) dx 给出。

A = ∫₀W d(x) dx

For a quadratic depth function, this becomes A = ∫₀W (ax² + bx + c) dx = [a x³/3 + b x²/2 + cx]₀W.

对于二次深度函数，此式变为 A = ∫₀W (ax² + bx + c) dx = [a x³/3 + b x²/2 + cx]₀W。

4. Volume of Growing Medium | 生长介质体积

nt, this reduces to V = LWd. If the depth varies according to d(x), the volume is V = L ∫₀W d(x) dx.

若屋顶长度为 L，基质总体积为 V = L × A。如果深度恒定，则简化为 V = LWd。如果深度按 d(x) 变化，则体积为 V = L ∫₀W d(x) dx。

V = L ∫₀W (ax² + bx + c) dx

example, if d(x) = kx(W - x), where k is a positive constant, then the volume is V = Lk ∫₀W (Wx - x²) dx = LkW³/6.

例如，如果 d(x) = kx(W - x)，其中 k 为正数，则体积为 V = Lk ∫₀W (Wx - x²) dx = LkW³/6。

5. Optimising Volume with Differentiation | 用微分法优化体积

e maximum depth, we differentiate d(x) with respect to x and set the derivative equal to zero.

假设深度由 d(x) = kx(W - x) 给出，其中 k > 0 且 0 ≤ x ≤ W。为了求最大深度，我们对 d(x) 关于 x 求导并令导数为零。

d'(x) = k(W - 2x) = 0 ⇒ x = W/2

The maximum depth occurs at the centre of the roof. Substituting x = W/2 gives dmax = k(W/2)(W/2) = kW²/4. The volume is then V = LkW³/6.

最大深度出现在屋顶中心。代入 x = W/2 得 dmax = k(W/2)(W/2) = kW²/4。体积则为 V = LkW³/6。

6. Surface Runoff and Exponential Decay | 地表径流与指数衰减

olume is V₀ and the decay constant is λ, then the retained volume at time t is V(t) = V₀ e−λt.

降雨后，浅绿层保留的水量因排水和蒸发而呈指数衰减。若初始保留体积为 V₀，衰减常数为 λ，则 t 时刻的保留体积为 V(t) = V₀ e−λt。

V(t) = V₀ e−λt

the retained volume to halve, is found by setting V(t) = V₀/2. This gives t₁/₂ = ln 2 / λ.

半衰期，即保留水量减半所需的时间，通过令 V(t) = V₀/2 求得。由此得到 t₁/₂ = ln 2 / λ。

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7. Geometric Constraints | 几何约束

The shallow green design must satisfy a maximum depth constraint, often set by structural load limits or plant root requirements. For example, if the maximum allowed depth is 0.15 m, then dmax ≤ 0.15. Using dmax = kW²/4, we obtain kW²/4 ≤ 0.15, which rearranges to k ≤ 0.6 / W².

浅绿设计必须满足最大深度限制，该限制通常由结构荷载限值或植物根系需求决定。例如，如果最大允许深度为 0.15 米，则 dmax ≤ 0.15。利用 dmax = kW²/4，可得 kW²/4 ≤ 0.15，整理得 k ≤ 0.6 / W²。

kW²/4 ≤ 0.15 ⇒ k ≤ 0.6 / W²

These inequalities link the parameter k with the roof width W, and they are often used to determine the feasible range of W for a given k.

这些不等式将参数 k 与屋顶宽度 W 联系在一起，通常用于确定给定 k 时 W 的可行范围。

8. Integration in Drainage Design | 排水设计中的积分

assumption leads to the differential equation dV/dt = −λV, where λ is a positive constant.

浅绿层的排水速率与保留水量成正比。这一物理假设导出微分方程 dV/dt = −λV，其中 λ 为正常数。

dV/dt = −λV

Solving by separation of variables gives V(t) = V₀ e−λt. This is exactly the exponential decay model used in the previous section, and it can be applied to estimate how quickly the roof dries after a storm.

通过分离变量法求解得到 V(t) = V₀ e−λt。这正是上一节使用的指数衰减模型，可用于估算暴雨后屋顶排干的速度。

9. Cost Optimisation | 成本优化

it costs are c₁ per cubic metre and c₂ per square metre, the total cost is C = c₁V + c₂LW.

基质成本与其体积成正比，防水层成本与屋顶面积成正比。如果单位成本分别为每立方米 c₁ 和每平方米 c₂，则总成本为 C = c₁V + c₂LW。

C = c₁V + c₂LW

and k are all positive, this derivative is always positive, so the cost increases with W. The optimal width is therefore the smallest width allowed by the depth and volume constraints.

利用 V = LkW³/6 并将 L 视为固定值，我们可以对 W 求导：dC/dW = c₁LkW²/2 + c₂L。如果 c₁、c₂ 和 k 均为正数，该导数始终为正，因此成本随 W 增大而增加。所以最优宽度是深度和体积约束允许的最小宽度。

10. Exam-Style Practice | 考试风格练习

Show that the maximum depth is 0.4 m and find the volume of the substrate. Then find the half-life if the retained water decays at a rate of 3% per hour.

一道典型的 Edexcel 题目可能给出 d(x) = 0.1x(4 - x) 且 0 ≤ x ≤ 4，L = 10。证明最大深度为 0.4 米并求基质体积。如果保留水量以每小时 3% 的速率衰减，求半衰期。

Differentiating gives d'(x) = 0.1(4 - 2x) = 0 at x = 2, so dmax = 0.1 × 2 × 2 = 0.4 m. The volume is V = 10 ∫₀⁴ 0.1x(4 - x) dx = [2x² - x³/3]₀⁴ = 32/3 m³.

求导得 d'(x) = 0.1(4 - 2x) = 0，当 x = 2 时取得最大值，所以 dmax = 0.1 × 2 × 2 = 0.4 米。体积为 V = 10 ∫₀⁴ 0.1x(4 - x) dx = [2x² - x³/3]₀⁴ = 32/3 立方米。

If the decay rate is 3% per hour, then λ = 0.03. The half-life is t₁/₂ = ln 2 / 0.03 ≈ 23.1 hours.

如果衰减率为每小时 3%，则 λ = 0.03。半衰期为 t₁/₂ = ln 2 / 0.03 ≈ 23.1 小时。

11. Common Mistakes | 常见错误

Students often forget to convert all measurements to the same unit before integrating. They may also use the wrong limits of integration or differentiate when they should integrate. Another frequent error is confusing exponential decay with linear decay.

学生经常忘记在积分前将所有测量值统一到相同单位。他们还可能使用错误的积分上下限，或在该积分时进行微分。另一个常见错误是将指数衰减与线性衰减混淆。