Related Rates

Unit 4 · Contextual Applications of Differentiation

Related Rates

In related rates problems, two or more quantities change with respect to time. Differentiate a geometric or algebraic relationship with respect to t using the Chain Rule, then solve for the unknown rate. Key rule: substitute specific values only after differentiating.

Motion Relationships

Position → Velocity → Acceleration

v(t) = s'(t) \qquad a(t) = v'(t) = s''(t)

Speed = |v(t)|; speeding up when v and a have the same sign

Common Geometric Relations

A_{\text{circle}} = \pi r^2 \implies \frac{dA}{dt} = 2\pi r\frac{dr}{dt}

V_{\text{cone}} = \frac{1}{3}\pi r^2 h \quad (\text{use similar triangles to eliminate } r)

AP Tip: Draw a diagram, label variables, write the geometric relation, differentiate with respect to t, then substitute known values.

Caution: Never substitute a specific value of a variable before differentiating — doing so eliminates its rate from the equation.

Type 1

Interpreting Rates and Particle Motion

Translate derivative values into real-world meaning: sign indicates direction (increase/decrease), magnitude gives the speed of change.

Example 1

A particle's position is s(t) = t³ − 9t² + 24t (meters, t ≥ 0 seconds). Find all times when the particle is at rest, and determine the direction of motion on (0, 1).

Example 2

v(t) = t² − 4t + 3 m/s. Is the particle speeding up or slowing down at t = 3?

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Type 2

Area and Volume Related Rates

Write a formula relating two quantities, differentiate with respect to t, then substitute known values of the rates and variables.

Example 3

A circular ripple expands at dr/dt = 3 ft/s. How fast is the area increasing when r = 8 ft?

Example 4

A conical tank (height 12 ft, top radius 4 ft) drains at 2 ft³/min. Find dh/dt when h = 6 ft.

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Type 3

Pythagorean (Ladder) Related Rates

Use the Pythagorean theorem to relate sides of a right triangle, then differentiate with respect to t.

Example 5

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

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