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Grade 12 Math Lab: Analysis & Inference FREE
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Related Rates
Learning Objectives
- Set up related rates problems
- Differentiate with respect to time
- Solve for unknown rates
Key Concepts
Related Rates
Finding how rates of change of related quantities are connected
Implicit Time Derivative
Differentiate with respect to t using chain rule
Theory
**Steps:**
1. Draw diagram, identify variables
2. Write equation relating variables
3. Differentiate both sides with respect to time t
4. Substitute known values and solve
**Common setups:**
- Expanding/shrinking shapes
- Ladder sliding down wall
- Filling/draining tanks
- Shadow problems
**Example:** V = πr²h → dV/dt = π(2r·dr/dt·h + r²·dh/dt)
Examples
Sphere radius grows at 2 cm/s. Rate of volume change when r=5?
Solution: V = 4πr³/3, dV/dt = 4πr²·dr/dt = 4π(25)(2) = 200π cm³/s