4.5 Solving Related Rates Problems

What are related rates?

Related rates problems involve two or more quantities that change over time and are linked by an equation. By differentiating the equation with respect to time (t) using implicit differentiation, you can relate the unknown rate to known rates. Every rate is a derivative like dx/dt or dV/dt.

The solving process

1) Draw a diagram and label variables. 2) Write an equation relating the variables (e.g., geometry formulas, Pythagorean theorem). 3) Differentiate both sides with respect to t, using the chain rule. 4) Substitute known values (rates and quantities) at the specific instant. 5) Solve for the unknown rate. Always plug in numbers AFTER differentiating.

Common setups

Classic scenarios include a ladder sliding down a wall, a shadow's length changing, balloons/spheres with changing volume, and cones filling with liquid. Watch units and signs: rates are positive if increasing, negative if decreasing. Keep constants (like a ladder's fixed length) as constants — do not differentiate them as variables.

Key terms

Related rates

Problems involving two or more variables that change with time, linked by an equation, solved using implicit differentiation with respect to t.

Key step before differentiating

Write an equation relating all relevant variables (e.g., geometric formula or Pythagorean theorem).

When to substitute known values

Only after differentiating the equation with respect to time — never before.

Sphere volume rate formula

dV/dt = 4πr² (dr/dt), from differentiating V = (4/3)πr³.

Ladder problem setup

Use x² + y² = L² (Pythagorean theorem) where L is constant, x is the base distance, and y is the height on the wall.

Try an AP-style question

A 13-ft ladder leans on a wall. Its base slides away at 2 ft/s. How fast is the top sliding down when the base is 5 ft from the wall?

1. 2 ft/s
2. 5/6 ft/s
3. 12/5 ft/s
4. 10/13 ft/s