Edexcel Maths 3.1 Exchange Surfaces | Edexcel 数学 3.1 交换表面

📚 Edexcel Maths 3.1 Exchange Surfaces | Edexcel 数学 3.1 交换表面

Exchange surfaces are usually introduced in biology, but in Edexcel Maths they also form a powerful modelling context. In this unit, you use surface area, volume, ratio, differentiation and related rates to analyse how efficiently a structure can exchange materials with its surroundings.

交换表面通常是在生物中引入的，但在 Edexcel 数学中，它们也构成了一个强大的建模情境。在本单元中，你将使用表面积、体积、比值、微分和相关变化率来分析一个结构与其周围环境交换物质的效率。

You will work with cubes, cylinders and spheres as simplified models of exchange surfaces. The key mathematical idea is that as a shape grows, its volume increases more quickly than its surface area, so the surface area to volume ratio falls.

你将使用立方体、圆柱体和球体作为交换表面的简化模型。关键的数学思想是：随着形状增大，其体积比表面积增长得更快，因此表面积与体积比会下降。

1. Surface Area to Volume Ratio (SA:V) | 表面积与体积比

In any exchange system, the key quantity is the surface area to volume ratio, often written as SA:V. For a cube with side length x, the surface area S is 6x² and the volume V is x³, so the ratio is 6/x. This shows that smaller cubes have a larger SA:V ratio than larger cubes.

在任何交换系统中，关键量是表面积与体积比，通常写作 SA:V。对于边长为 x 的立方体，表面积 S 为 6x²，体积 V 为 x³，因此比值为 6/x。这表明较小立方体的 SA:V 比值大于较大立方体。

bically, so the ratio decreases. This is why large organisms need specialised exchange surfaces such as lungs or gills rather than relying on simple diffusion across the skin.

随着 x 增大，分子 6x² 呈二次增长，而 V = x³ 呈三次增长，因此比值下降。这就是为什么大型生物需要肺或鳃等特化交换表面，而不能仅依赖皮肤进行简单扩散。

Cube: S = 6x², V = x³, SA:V = 6/x

2. Calculating Surface Areas of Common Solids | 常见几何体的表面积计算

To model exchange surfaces, you need to calculate surface areas and volumes of common solids. The table below shows the formulas for a cube, cuboid, sphere and cylinder.

为了对交换表面进行建模，你需要计算常见几何体的表面积和体积。下表给出了立方体、长方体、球体和圆柱体的公式。

These formulas are essential because many exchange structures can be approximated by these shapes. Always identify which solid best matches the model before substituting values.

这些公式非常重要，因为许多交换结构都可以用这些形状近似。在代入数值之前，一定要先确定哪个几何体最符合模型。

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3. Volume Calculations and Units | 体积计算与单位

e in cm² and volumes in cm³. Conversions include 1 cm³ = 1 × 10⁻⁶ m³ and 1 dm³ = 1000 cm³.

在 Edexcel 数学问题中，单位必须仔细处理。如果长度单位为 cm，则面积单位为 cm²，体积单位为 cm³。换算包括 1 cm³ = 1 × 10⁻⁶ m³，以及 1 dm³ = 1000 cm³。

Always convert all lengths to the same unit before substituting into formulas. Common mistakes include mixing mm, cm and m in the same calculation or forgetting that area and volume conversions scale by the square and cube of the length factor.

在代入公式之前，一定要将所有长度转换为相同单位。常见错误包括在同一计算中混用 mm、cm 和 m，或者忘记面积和体积换算要按长度因子的平方和立方进行缩放。

• 1 m = 100 cm, so 1 m² = 10 000 cm²
• 1 m = 100 cm, so 1 m³ = 1 000 000 cm³
• 1 cm³ = 0.001 dm³ = 1 mL

1 m = 100 cm，因此 1 m² = 10 000 cm²；1 m³ = 1 000 000 cm³；1 cm³ = 0.001 dm³ = 1 mL。

4. Modelling Diffusion Rate with Fick's Law | 用菲克定律建模扩散速率

is surface area, ΔC is the concentration difference, d is the diffusion distance and D is the diffusion coefficient.

对于数学建模，交换速率 R 可以用简化的菲克型定律描述：R = DA ΔC / d，其中 A 是表面积，ΔC 是浓度差，d 是扩散距离，D 是扩散系数。

This gives direct proportion: R ∝ A and R ∝ ΔC, but inverse proportion R ∝ 1/d. Doubling surface area doubles the rate; doubling diffusion distance halves the rate.

这给出了正比例关系：R ∝ A 和 R ∝ ΔC，但反比例关系 R ∝ 1/d。表面积加倍则速率加倍；扩散距离加倍则速率减半。

R = DA ΔC / d

You may be asked to predict the effect of changing one variable while keeping the others constant, or to rearrange the formula to make A or d the subject.

考试中可能会要求你预测在保持其他变量不变的情况下改变一个变量的影响，或者将公式变形，以 A 或 d 作为主项。

5. Exchange Efficiency and SA:V Ratio | 交换效率与 SA:V 比

Exchange efficiency is often measured by SA:V. Consider two cubes: one with side 1 cm and one with side 3 cm.

交换效率通常用 SA:V 来衡量。考虑两个立方体：一个边长为 1 cm，另一个边长为 3 cm。

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