Lesson 5: 4-5 Solving Related Rates
Duration of Days: 2
Lesson Objective
Use Similar Triangles to establish relationships between variables in a system before differentiating (essential for conical tanks and shadow problems).
Apply the Pythagorean Theorem as the primary equation for motion problems involving right angles (e.g., ships moving in different directions or ladders).
Use Trigonometric Ratios (sin, cos, tan) to relate an angle of elevation/depression to the rate of change of a side length.
Solve "In/Out" related rates problems where a volume is changing based on a net rate of flow.
Why is it necessary to use similar triangles to write the radius r in terms of height h before differentiating a cone's volume?
How does the rate of change of an angle behave as an object gets closer to or further from an observer?
In a "ladder" problem, why is the rate of change of the length of the ladder always zero?
How can we use the Product Rule or Quotient Rule effectively when multiple parts of a geometric formula are changing over time?
Similar Triangles
Pythagorean Theorem
Angle of Elevation
Angle of Depression
Constant of Proportionality (in cones)
Related Angles
Hypotenuse
Topic 4.5: Solving Related Rates Problems.
Standard CHA-3.E: Solve problems involving related rates.
Mathematical Practice 1.E: Apply appropriate mathematical rules or procedures, including the Chain Rule for implicit differentiation.
Description:
This lesson focuses on "multi-step" modeling. Students learn that they often have "too many variables" in their primary equation. They must find a secondary relationship to substitute one variable out before they differentiate. This section also introduces problems involving the rate of change of an angle.
Purpose: This is the most frequent way Related Rates appear on the Free Response section of the AP Exam. Students must be able to handle "Constraint Equations" (like the ratio of a cone's dimensions) to reduce the complexity of the calculus.
DOK 2 (Skill/Concept): Differentiating the Pythagorean theorem to find how fast the distance between two moving cars is changing.
DOK 3 (Strategic Thinking): Setting up the relationship for a shadow problem, identifying that the tip of the shadow and the length of the shadow change at different rates.
DOK 4 (Extended Thinking): Solving a conical tank problem where water is entering at one rate and leaking at another, requiring a combination of Net Rate concepts (from 4.3) and Similar Triangles (from 4.5).
Scaffolding (Support):
The "Zero Rate" Rule: Remind students to identify parts of the problem that are "frozen" (e.g., the height of a lamp post, the length of a ladder). If a value is constant, its derivative is 0.
Triangle Cheat Sheet: Provide a visual guide showing when to use Pythagorean Theorem (finding a side) vs. Trig (finding an angle) vs. Similar Triangles (reducing variables).
Extension (Challenge):The "Three-Way" Rate: A problem where three sides of a triangle are all changing at different rates (e.g., two people running at an angle to each other).
Inverse Trig Related Rates: Finding the rate of change of an angle by differentiating theta = arctan(y/x).
College Board AP Classroom Assessments