Connected Rates of Change

Connected-rates problems link quantities that vary with the same independent variable, usually time.

Method

1. Identify the rate required and introduce the common independent variable: for example, \(\dfrac{\mathrm{d}V}{\mathrm{d}t}\).
2. Identify the given rate, for example \(\dfrac{\mathrm{d}A}{\mathrm{d}t}\).
3. Apply the chain rule: \(\dfrac{\mathrm{d}V}{\mathrm{d}t}=\dfrac{\mathrm{d}V}{\mathrm{d}A}\dfrac{\mathrm{d}A}{\mathrm{d}t}\).
4. Use the geometry or physical condition to express \(V\) in terms of \(A\), then find \(\dfrac{\mathrm{d}V}{\mathrm{d}A}\).
5. If a direct relation is inconvenient, connect both quantities through another variable such as \(r\): \(\dfrac{\mathrm{d}V}{\mathrm{d}A}=\dfrac{\mathrm{d}V}{\mathrm{d}r}\dfrac{\mathrm{d}r}{\mathrm{d}A}\).
6. Substitute the relevant instantaneous condition only after differentiating, then include the correct units and sign.