AP Calculus AB Lesson

Related Rates: When Everything Changes at Once

Shared by A STEM Educator

11th GradeThin-SlicingCHA-3.D, CHA-3.E50 min

Learning Objective

Students will be able to identify related rates in geometric and physical contexts, set up implicit differentiation equations relating multiple rates, and solve for unknown rates — including multi-variable scenarios involving cones, ladders, and moving shadows.

Lesson Overview

Students progress from recognizing rates in familiar derivative notation to setting up and solving multi-variable related rates problems. The lesson builds from implicit differentiation of simple geometric formulas through the classic ladder problem to a shadow-length problem combining two changing quantities.

Materials

Vertical whiteboards or chart paper
Dry-erase markers
Markers in 2 colors per group
Random grouping tool
Rulers (for diagram tasks)

Scaffolded Task Progression

1Task 1

A square's side length s is increasing at 4 cm/s. Write an equation for the area A in terms of s. Then differentiate both sides with respect to time t to find dA/dt. When s = 3 cm, how fast is the area increasing?

[ Interactive graph / diagram — available in the full lesson ]

2Task 2

A spherical balloon is being deflated. Its radius is decreasing at 0.5 cm/s. When r = 8 cm, how fast is the volume decreasing?

Before writing any equation: Which rate is negative in this problem, and WHY? Write a one-sentence explanation on your whiteboard before doing any calculations.

Unlock the full lesson

See all 7 scaffolded tasks, the full answer key, warm-up, exit slip, and a print-ready version — free to start.

Create a free account

Already have an account? Sign in

Make it your own

Like this lesson but want to tweak it?

Make an editable copy and adjust the tasks, warm-up, and answer key to fit your class — no need to start from scratch.

Generate a new lesson