📚 Example 5.2.2: Connected Rates of Change | 示例5.2.2：相关变化率
Connected rates of change are a central application of differentiation in AQA A-Level Mathematics. They appear when two or more variables depend on time and are linked by a geometric or physical relationship. By using the chain rule, you can relate one rate, such as dV/dt, to another, such as dr/dt.
相关变化率是 AQA A-Level 数学中导数的核心应用之一。当两个或多个变量随时间变化并通过几何或物理关系相互联系时，就会出现此类问题。利用链式法则，你可以将一个变化率（如 dV/dt）与另一个变化率（如 dr/dt）联系起来。
Typical exam questions involve a balloon being inflated, a tank being filled, or a ladder sliding down a wall. In each case, you are given one rate of change and must find another. The solution almost always follows the same logical structure, so mastery of this topic gives you a reliable set of marks.
典型的考试题目包括气球充气、水箱注水或梯子滑下墙壁。在每种情况下，题目会给出一个变化速率，要求求出另一个变化速率。解题过程几乎总是遵循相同的逻辑结构，因此掌握这一主题能为你提供稳定的得分点。
1. What Are Connected Rates of Change? | 什么是相关变化率？
In a modelling or geometry problem, a variable such as volume V, radius r, area A, or height h may depend on time t. If two of these quantities are related by an equation, their rates of change with respect to time are connected through differentiation.
在建模或几何问题中，体积 V、半径 r、面积 A 或高度 h 等变量可能随时间 t 变化。如果其中两个量通过方程关联，那么它们对时间的变化率就会通过导数相互联系。
You are usually given the rate of change of one quantity and asked to find the rate of change of another. The key idea is not to differentiate with respect to t directly, but to use an intermediate variable such as r or x. This is where the chain rule becomes essential.
题目通常会给出一个量的变化速率，要求求出另一个量的变化速率。关键思想不是直接对 t 求导，而是使用中间变量，例如 r 或 x。这正是链式法则不可或缺的地方。
2. The Chain Rule as the Key Tool | 链式法则是核心工具
For a function where V depends on r, and r depends on t, the chain rule gives:
若 V 依赖于 r，而 r 依赖于 t，链式法则给出：
dV/dt = (dV/dr) × (dr/dt)
This formula lets you find one unknown rate if the other two parts are known. In AQA exam questions, identifying this structure is often the first mark, so always write it down clearly before substituting values.
该公式使你能在已知另外两部分的情况下求出一个未知变化率。在 AQA 考试题中，识别这一结构通常就是第一个得分点，因此在代入数值之前务必清晰地写出该公式。
se the intermediate variable that makes the differentiation straightforward and matches the given information.
同样的模式适用于任何链式关系，例如 dA/dt = (dA/dr)(dr/dt) 或 dy/dt = (dy/dx)(dx/dt)。你应选择一个中间变量，使求导过程简洁并与已知信息相匹配。
3. Step-by-Step Method | 分步解题法
Follow these steps to solve a connected rates problem accurately:
按照以下步骤可准确求解相关变化率问题：
- Write down the formula linking the two variables, such as V = (4/3)πr³. — 写出联系两个变量的公式，例如 V = (4/3)πr³。
- Differentiate with respect to the intermediate variable, not t. — 对中间变量求导，而不是对 t 求导。
- Write down the chain rule dV/dt = (dV/dr)(dr/dt). — 写出链式法则 dV/dt = (dV/dr)(dr/dt)。
- Substitute the known rates and the value of the intermediate variable. — 代入已知变化率和中间变量的值。
- Solve for the required rate and include correct units. — 解出所求变化率并写出正确单位。
This method is robust and helps you avoid common mistakes such as substituting numbers before differentiating. Premature substitution can make the derivative harder to compute and often leads to losing the structure of the chain rule.
这一方法稳健可靠，能帮助你避免在求导前就代入数字等常见错误。过早代入数值会使求导变得更加困难，并常常导致丢失链式法则的结构。
4. Worked Example: Inflating Spherical Balloon | 例题：充气中的球形气球
A spherical balloon is inflated at a constant rate of 12 cm³ s⁻¹. Find the rate at which the radius increases when the radius is 5 cm.
一个球形气球以 12 cm³ s⁻¹ 的恒定速率充气。求当半径为 5 cm 时半径增加的速率。
The volume of a sphere is V = (4/3)πr³, so differentiating with respect to r gives:
球体积公式为 V = (4/3)πr³，因此对 r 求导得：
dV/dr = 4πr²
Using the chain rule:
使用链式法则：
dV/dt = (dV/dr) × (dr/dt)
Substitute dV/dt = 12 and dV/dr = 4πr²:
代入 dV/dt = 12 和 dV/dr = 4πr²：
12 = 4πr² × (dr/dt)
Rearrange and set r = 5:
移项并令 r = 5：
dr/dt = 12 / (4π × 5²) = 3 / (25π) ≈ 0.0382 cm s⁻¹
The radius is increasing at approximately 0.0382 cm s⁻¹, which is about 0.4 mm per second. This answer is exact if written as 3/(25π) cm s⁻¹, and it is often better to leave it in this form unless the question asks for a decimal.
半径以约 0.0382 cm s⁻¹ 的速率增加，大约为每秒 0.4 毫米。若答案写成 3/(25π) cm s⁻¹ 则为精确值，除非题目要求小数，否则通常最好保留这种形式。
5. Interpreting the Sign of the Rate | 变化率正负号的解释
A positive rate means the quantity is increasing over time. A negative rate means it is decreasing. In balloon deflation, dV/dt would be negative, so dr/dt would also be negative because 4πr² is always positive.
正的变化率表示该量随时间增加。负变化率表示该量随时间减少。在气球放气时，dV/dt 为负，因此 dr/dt 也为负，因为 4πr² 始终为正。
Always check whether your final sign matches the physical situation. For example, if water leaks out of
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