Related rates – sliding ladder, filling cone, ripple, balloon and shadow
MathematicsCalculusAges 16–17
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Sign in to playFive animated scenarios: a ladder sliding down a wall, water pouring into a conical tank, a circular ripple on a pond, a balloon being inflated and a person walking away from a street lamp. Set one rate (dx/dt, dV/dt or dr/dt) and press Play: linked gauges and graphs against time show how the other rates change, and the chain-rule steps – differentiating the relation with respect to t – are filled in with the numbers at the current instant.
Lesson: Related rates: differentiating a relation with respect to time using the chain rule
What it shows
In a related-rates problem one quantity changes at a known rate and we want the rate of another quantity linked to it. Write the relation (Pythagoras for the ladder, V = πR²h³/(3H²) for the cone, A = πr², V = 4πr³/3, similar triangles for the shadow), differentiate both sides with respect to time t using the chain rule, then substitute the values at the instant in question. The simulation animates each case and checks the answer with Δ/Δt over 0.001 s. See also Motion on a line with calculus and Real-world optimization with derivatives.
How to use
Choose a Scenario, set the given rate and the sizes with the sliders, then press Play or drag the Instant slider to pick a moment. Read the three gauges (the given rate and two derived rates), the graphs of the quantity and its rate against t, and the numbered steps underneath. Tick Check with Δ/Δt to compare with a finite difference, and press Reset to start again.
Parameters you can change
- Scenario Ladder sliding down a wall, Filling a conical tank, Ripple spreading on a pond, Inflating a balloon, Walking away from a lamp
- Ladder length L 3–12 m
- Rate at which the foot slides out, dx/dt 0.1–2 m/s
- Radius of the tank top R 4–15 cm
- Height of the tank H 10–30 cm
- Flow rate dV/dt (negative = draining) -100–100 cm³/s
- Rate at which the ripple spreads, dr/dt 0.1–2 m/s
- Rate of inflation dV/dt 20–500 cm³/s
- Height of the lamp post H 3–10 m
- Height of the person h 1–2.2 m
- Walking speed dx/dt 0.2–3 m/s
- Show the Δ/Δt check
- Play automatically on opening
Questions to explore
- The foot of the ladder slides out at a constant rate, so why does the top fall faster and faster?
- Water flows into the cone at a constant rate; how does dh/dt change when the depth doubles?
- Why does the tip of the shadow move at the same speed whether the person is near the lamp or far away?
