Related Rates: Draining Conical Tank

Drain Setup

Drain rate dV/dt 0.60

Tank taper R / H 0.60

Live Readouts

Water height h

3.00

Elapsed time t

0.0 s

Remaining volume V

—

Surface radius r

—

Instantaneous dh/dt (related rate)

—

V = (πR²/3H²)·h³
dV/dt = (πR²/H²)·h²·dh/dt

How it works

The tank is a cone of fixed height H and rim radius R, apex down. Because the water's surface radius r grows in proportion to its height h (similar triangles: r/h = R/H), its volume as a function of height alone is

r = (R/H)·h
V(h) = (1/3)·π·r²·h = (πR² / 3H²)·h³

The drain removes volume at a chosen constant rate, dV/dt = −k. Differentiating V(h(t)) with respect to time using the chain rule (implicit differentiation — this is the textbook "related rates" technique) gives:

dV/dt = dV/dh · dh/dt = (πR²/H²)·h²·dh/dt
        ⇒  dh/dt = (dV/dt) / [(πR²/H²)·h²]

Even though dV/dt is held constant by the slider, dh/dt is not constant — it depends on h² in the denominator, so the water level falls slowly while the tank is full (large h, wide cross-section) and accelerates sharply as the tank empties (h → 0, cross-section → a point). That non-linear speed-up is exactly what the falling water surface and the live dh/dt readout demonstrate.

Drain rate dV/dt — the constant volumetric outflow through the apex; the simulator integrates V(t) = V₀ − k·t directly, then recovers h(t) by inverting V(h).
Tank taper R/H — a narrower cone (small R/H) makes dh/dt swing more violently near empty, since the h² term shrinks faster relative to volume.
dh/dt readout — evaluated analytically from the related-rates formula above at the current h, not by finite-differencing the animation, so it is the exact calculus answer at every instant.

This is the classic "draining/filling conical tank" related-rates problem from a first calculus course, generalized so any drain rate and any cone shape can be explored instead of a single textbook number.

Water surface Tank wall Draining stream

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