Related Rates

The five-step approach

Every related rates problem follows the same template:

1. Identify variables. What quantities are changing? Which rates are given? Which is unknown?
2. Write a relating equation. Geometry, physics, or definition that connects the changing quantities.
3. Differentiate with respect to t. Apply the chain rule — every spatial variable becomes a function of time.
4. Plug in given values. Substitute the values of the variables AND given rates at the moment of interest.
5. Solve for the unknown rate.

Critical — perform step 3 (differentiate) BEFORE step 4 (plug in). Plugging in numerical values first turns variables into constants, whose derivative is zero — destroying the calculation.

Example 1 — sliding ladder

A 13-foot ladder leans against a wall. The bottom slides away from the wall at 2 ft/sec. How fast is the top sliding down when the bottom is 5 feet from the wall?

Variables. x = distance from wall to ladder bottom; y = height of ladder top on wall.

Relating equation. By Pythagoras, x² + y² = 13² = 169.

Differentiate with respect to t. Both x and y are functions of t:

2x · (dx/dt) + 2y · (dy/dt) = 0

Plug in. At the moment of interest, x = 5. From x² + y² = 169 — y² = 144 — y = 12. Given dx/dt = 2 ft/sec.

2(5)(2) + 2(12)(dy/dt) = 0
20 + 24(dy/dt) = 0
dy/dt = −20/24 = −5/6 ft/sec

Negative because y is decreasing. The top is sliding down at 5/6 ft/sec ≈ 0.83 ft/sec.

Example 2 — inflating balloon

A spherical balloon is inflated at 100 cm³/sec. How fast is the radius increasing when the radius is 10 cm?

Variables. V = volume, r = radius, both functions of t.

Relating equation. V = (4/3)πr³.

Differentiate.

dV/dt = 4πr² · (dr/dt)

Plug in. At r = 10, dV/dt = 100:

100 = 4π(10)²·(dr/dt) = 400π·(dr/dt)
dr/dt = 100/(400π) = 1/(4π) ≈ 0.0796 cm/sec

The radius increases by ~0.08 cm/sec at that moment. Note — as the balloon gets larger (larger r²), the same volume rate produces slower radius growth (dr/dt decreases). Inflating "feels slower" near full size.

Example 3 — shadow lengthening

A 6-foot tall person walks away from a 15-foot streetlight at 5 ft/sec. How fast is the tip of their shadow moving?

Variables. x = distance from light to person; s = length of shadow; T = position of shadow tip = x + s.

Relating equation. Similar triangles — light/total height = person/shadow.

15 / (x + s) = 6 / s
15s = 6(x + s) = 6x + 6s
9s = 6x
s = (2/3)x

So shadow tip position T = x + s = x + (2/3)x = (5/3)x.

Differentiate.

dT/dt = (5/3) · dx/dt = (5/3)(5) = 25/3 ≈ 8.33 ft/sec

The shadow tip moves at 25/3 ft/sec — faster than the person walks. This is independent of how far the person has walked — the shadow tip's speed is a fixed multiple of the walking speed, determined only by the geometry.

Common pattern types

Pattern	Relating equation	What you typically solve
Right-triangle / Pythagoras	x² + y² = constant	One side from rate of another
Volume of geometric shape	V = formula(dimensions)	Dimension rate from volume rate or vice versa
Similar triangles	Ratio of similar dimensions	Distant length from near rate
Trigonometric (angle of elevation)	tan θ = height/distance	Angle rate from distance rate
Conical tank	V = (1/3)πr²h with r/h = constant	Height rate from fill rate
Position with time	distance = √((Δx)² + (Δy)²)	Closing/separating rate of two objects

JavaScript — verification of related rates problems

// Verify example 1 numerically
function ladderProblem() {
  const L = 13;       // ladder length
  const dxdt = 2;     // ft/sec
  const x_now = 5;
  const y_now = Math.sqrt(L*L - x_now*x_now);

  // Symbolic answer
  const dydt = -(x_now * dxdt) / y_now;
  console.log('dy/dt =', dydt);  // -5/6 ≈ -0.833

  // Numerical verification — simulate forward
  const dt = 0.001;
  const x_after = x_now + dxdt * dt;
  const y_after = Math.sqrt(L*L - x_after*x_after);
  const dydt_numerical = (y_after - y_now) / dt;
  console.log('Numerical:', dydt_numerical);  // ≈ -0.833

  // They should agree (modulo dt-sized error)
}

ladderProblem();

Where related rates show up

• Physics — kinematics. Velocity is dx/dt. Acceleration is dv/dt. Rotational angular velocity. Trajectory analysis. The whole subject is related rates with constraint equations.
• Engineering — fluid dynamics. Tank filling, draining, pressure changes related to flow rates and tank geometry.
• Astronomy — orbital mechanics. Apparent motion of planets, conjunction events, transit timing.
• Finance — derivatives (in the financial sense). Rate of change of option prices with respect to underlying. The "Greeks" (delta, gamma, theta, vega, rho) are partial derivatives of options.
• Pharmacology. Drug concentration in blood — IV administration rate, clearance rate, distribution into tissues. All chain-rule-with-time problems.
• Computer graphics — animation. When a model deforms, related rates between mesh vertices, normals, and rendered pixels — all coupled by chain rule.

Common mistakes

• Plugging in values before differentiating. The most common error. If x = 5, then x² = 25, but the derivative of 25 is 0 — destroying the calculation. Differentiate first (in symbols), then substitute values.
• Forgetting the chain rule. When y is a function of t, d/dt(y²) is 2y · dy/dt, not 2y. Skipping the chain rule factor produces wrong answers.
• Confusing the rate with the magnitude. dx/dt is the rate of change of x with respect to t. It can be negative (x decreasing) or vary. The numerical sign matters.
• Wrong units. Volume is cm³, rate is cm³/sec; radius is cm, rate is cm/sec. Mixing units gives meaningless answers. Track units throughout.
• Forgetting that time is implicit. Every spatial variable is a function of t — even if not written explicitly. d/dt(x²) = 2x · dx/dt, not just 2x.
• Solving the wrong unknown. Read the question carefully. Sometimes the question asks for dV/dt; sometimes for dr/dt. Don't compute one and report the other.