Related Rates

Related Rates: Finding How Quantities Change Together

Related rates problems arise whenever two or more quantities depend on each other and change over time. As a ladder slides down a wall, its height and distance from the wall both change. As water drains from a tank, its depth and volume both change. To solve these problems, we use an equation connecting the variables, differentiate it implicitly with respect to time $t$, and solve for the unknown rate.

The Core Strategy

Every related-rates problem follows four steps:

1. Draw and label — sketch the situation and assign variables to all changing quantities
2. Identify rates — write given and unknown rates as derivatives (e.g., $\frac{dx}{dt}$, $\frac{dV}{dt}$)
3. Find a relationship — write an equation connecting the variables using geometry, physics, or other constraints
4. Differentiate, then substitute — use implicit differentiation with respect to $t$, then plug in known numeric values and solve

The critical rule: differentiate first, substitute numbers second. Substituting a specific value before differentiating destroys the variable and gives zero for its derivative, which is incorrect.

Example: Sliding Ladder

A 10-ft ladder leans against a wall; its base slides away at $2\ \text{ft/s}$. How fast does the top fall when the base is 6 ft out?

With $x$ = distance from wall and $y$ = height on wall, the Pythagorean theorem gives: $x^2 + y^2 = 100$

Differentiate with respect to $t$: $2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 0$

When $x = 6$: $y = \sqrt{100 - 36} = 8$. Substituting $\frac{dx}{dt} = 2$: $2(6)(2) + 2(8)\frac{dy}{dt} = 0 \implies \frac{dy}{dt} = -\frac{3}{2}\ \text{ft/s}$

The negative sign indicates the top is descending.

Key Insight

Rates are signed quantities: positive means increasing, negative means decreasing. Always verify your answer's sign makes physical sense. Common tools include the Pythagorean theorem, area/volume formulas, and similar triangles—apply the same differentiate-then-substitute method regardless of the setup.