Related Rates: Four-Step Method & Oil-Spill Example

Context
- Go beyond single–variable differential calculus; deal with several inter-dependent quantities that evolve together over time.
- Main objective: determine an unknown rate of change using known rates + algebraic/physical relationships.
Typical wording
- Phrases like grows, shrinks, speeds up, slows down, how fast, how slowly ⇒ all signal derivatives.

Presence of two or more quantities that are function(s) of time t.
The problem supplies at least one numerical rate (e.g. \frac{dr}{dt}=2) and asks for another.
There is a static relationship (geometric, physical, empirical) tying the quantities together (e.g. A = \pi r^2 for a circle).

Step 1 – Identify Quantities & Rates
- List every variable that changes with time.
- Separate into:
- Known instantaneous values (e.g. r=5\text{ m} at a snapshot).
- Known instantaneous rates (e.g. \frac{dr}{dt}=2\text{ m/min}).
- Unknown rate(s) to solve for.
Step 2 – Establish Relationship(s)
- Could be
- Geometric (area–radius, volume–height, Pythagorean, etc.).
- Physical (Ideal Gas Law, Coulomb’s Law, Hooke’s Law).
- Economic/Natural (profit vs. tickets, concentration vs. volume).
- Write one (or several) equation(s) linking the variables before differentiating.
Step 3 – Differentiate Implicitly w.r.t. Time t
- Apply \frac{d}{dt} to both sides.
- Use Chain Rule whenever the variable itself depends on t.
- Outcome: an equation containing \frac{d(\text{quantity})}{dt} terms only.
Step 4 – Substitute Numerical Data & Solve
- Plug the snapshot values (quantities & known rates) into the differentiated equation.
- Algebraically isolate the unknown rate.
- Unit check (sanity test):
- Length ⇒ \text{m}, rate of length ⇒ \text{m}/\text{unit time}.
- Area ⇒ \text{m}^2, rate ⇒ \text{m}^2/\text{unit time}, etc.