Lesson 8: Related Rates of Change & Chain Rule Applications
Applying differentiation methods, rules, optimisation, and related rates for NCEA Level 3 Calculus.
🎯 Ngā Whāinga Akoranga | Learning Intentions
Related rates equations connecting multiple changing variables with respect to time (t) using the chain rule (e.g., dV/dt = (dV/dr) * (dr/dt)).
Formulate geometric relationships (spheres, cones, triangles), differentiate implicitly with respect to time t, and solve instantaneous rate of change problems.
🎥 Media Anchor & Pedagogical Scaffold
Related Rates — Khan Academy
Video (7 min 17 sec, Khan Academy): A comprehensive introduction to related rates problems, showing how to connect rates of change of different variables through geometric and implicit differentiation. Covers the classic water-tank and ladder examples.
🧠 1. Before Viewing (Activate & Predict)
If water is poured into a conical funnel at a constant rate, why does the water height rise more slowly as the cone fills up?
👁️ 2. During Viewing (Watch With a Job)
Watch the full 7-minute clip. In your calculus logbook, record:
- The related rates strategy: Write the key steps: (1) Draw and label the geometry; (2) identify given rates and target rate; (3) write a relationship equation (e.g., from Pythagorean theorem or V = πr²h); (4) differentiate both sides with respect to t; (5) substitute known values and solve for the unknown rate.
- A worked example: A 5-metre ladder leans against a wall. The base slides away from the wall at 2 m/s. When the base is 3 m from the wall, how fast is the top sliding down? (Pythagorean: x² + y² = 25; differentiate: 2x(dx/dt) + 2y(dy/dt) = 0; at x = 3, y = 4; so 6(2) + 8(dy/dt) = 0 → dy/dt = -1.5 m/s.)
- Chain rule link: Note how the chain rule appears naturally when differentiating by time.
🗣️ 3. After Viewing & Kaiako Move (Process & Apply)
Kaiako Move: Emphasise that given rates MUST be substituted ONLY AFTER differentiation, never before (common NCEA exam trap).
Immediate Task: Add to your Level 3 Calculus revision logbook (section 8): Related Rates Geometric Problems & Time Derivative Log.
⚡ Whakaoho | Do Now: Rates Already Linked (10 mins)
Write the volume of a sphere in terms of its radius. Differentiate it with respect to r. Now say in words what dV/dr means for a balloon, and how it differs from dV/dt. Today's whole method is the chain rule connecting those two.
📖 Activity 1: Chaining Two Rates Together (25 mins)
Use dV/dt = dV/dr · dr/dt throughout. A spherical balloon is inflated at 100 cm³/s: find how fast the radius grows when r = 5 cm. Then water drains from a cylindrical tank of radius 2 m at 0.5 m³/min: find how fast the depth falls. Before calculating either, write down which rate you are given and which you want. Most errors in this topic are solving for the wrong one, not differentiating incorrectly.
📝 Activity 2: Exam-Style Practice — Set Up, Then Solve (20 mins)
A ladder 5 m long slides down a wall with its foot moving out at 0.2 m/s. Find how fast the top descends when the foot is 3 m from the wall. For Merit, state the relationship between the variables before differentiating. For Excellence, explain why the top's speed increases as the foot moves further out, using your expression rather than intuition.
🏫 Kaiako Planning & Pedagogy Notes
NCEA Level 3 Alignment: Direct preparation for Level 3 Calculus (Apply differentiation methods in solving problems). Emphasise complete algebraic working and clear context conclusions for Merit/Excellence grades.