☰

Lesson 12: Related Rates – Classic Problems

1. Related Rates – Step-by-Step Method

  1. Draw a picture & define variables
  2. Write equation relating variables
  3. Differentiate both sides w.r.t. time \(t\)
  4. Plug in known values & solve for unknown rate

Example: Ladder sliding down wall

\(x^2 + y^2 = L^2\) (\(L\) fixed length)

\(2x dx/dt + 2y dy/dt = 0\)

\(dy/dt = -(x/y) dx/dt\)

Exercise 1

Ladder 13 ft long, base 5 ft from wall. Base moves away at 2 ft/s.
Find rate ladder top slides down when base is 12 ft out.
\(x = 12, y =\) , \(dx/dt = 2\)
\(dy/dt = -\)/\( · 2 = \) ft/s (downward)

2. Cone Water Tank Problem

Water poured into conical tank (radius \(r\), height \(h\), similar triangles).

V = \((1/3)\pi r^2 h\)

\(r/h =\) constant (say 2/5) → \(r = (2/5)h\)

\(V = (1/3)\pi (4/25 h^2) h = (4/75)\pi h^3\)

\(dV/dt = (4/75)\pi · 3 h^2 dh/dt\)

Exercise 2

Water rises at \(dh/dt = 0.1\) m/s in a cone. How fast is volume increasing when \(h = 5\) m (assume r = 0.4h)?

Exercise 3

Which problems typically use related rates? (Select all that apply)
← Previous Lesson (11) Next Lesson (13) →