Mathematics 11–12 · Year 12
Draining a funnel: related rates and Torricelli's law
Further applications of calculus: Further rates of change (Mathematics Extension 1, Year 12: the related-rates step through the chain rule); Applications of calculus: Rates of change (Mathematics Advanced, Year 12: reading dh/dt from the depth-time graph only, because related rates are not in the Mathematics Advanced syllabus)
This site has no interactive model of its own. Where a step or a material names a Concept Studio model, simulation or tool, it has not been built; an external simulation a step names (for example PhET) is not part of this site.
The idea
When water drains from a cone through a small hole, the depth falls at a rate linked to the volume rate by the chain rule, and the link changes as the surface shrinks.
What you need
- A plastic funnel of rim radius about 5 cm and depth about 10 cm, with its stem cut off where it meets the cone and the opening closed by a flat plastic disc sealed with waterproof tape, through which the teacher has drilled a 3 mm hole beforehand
- A ruler taped inside the funnel as a depth scale with 0 at the hole, a retort stand and clamp
- A stopwatch, water, a 500 mL beaker
How to do it
- Clamp the funnel over the beaker with a finger over the hole; fill to a depth of 8 cm above the hole.
- Release and read the depth every 5 s until empty; three runs.
- Plot depth against time and note where the depth falls fastest.
- Measure the rim radius r, the radius r0 of the cut end and the depth D from the hole to the rim: extending the cone's sides, its missing point lies h0 = r0 D / (r - r0) below the hole, and the full cone height is H = D + h0. With h measured from that point, write V = (pi/3)(r/H)^2 (h^3 - h0^3) and differentiate with the chain rule (Extension 1): dV/dt = pi (r/H)^2 h^2 dh/dt.
- Model the outflow with Torricelli's law dV/dt = -C a sqrt(2 g (h - h0)), where h - h0 is the depth of water above the hole; choose C so the model matches your first 10 s, then predict dh/dt when the water is 5 cm and 3 cm deep above the hole and compare with gradients read from your graph.
What you should see
Model check for an ideal cone that ends in a point at the hole (r = 5.0 cm, H = 10.0 cm, a 3.0 mm hole with a = 7.07 mm^2, C = 0.6 and g = 9.80 m/s^2): the outflow is 5.94 mL/s at h = 10 cm and the depth falls at 0.076 cm/s; at h = 8 cm the outflow is 5.31 mL/s and the depth falls at 0.106 cm/s; at h = 5 cm the outflow is only 4.20 mL/s but the depth falls at 0.214 cm/s, because the surface area has shrunk to a quarter. A real funnel is not pointed. Left on, a 9 cm stem would add its length to the head: with 5 cm of water in the cone the head becomes 14 cm and sqrt(14/5) = 1.67, so a plain orifice at the same discharge coefficient would pass 67 per cent more, 7.03 mL/s. Take that as an upper bound rather than a prediction, because 9 cm of 3 mm bore is a tube 30 diameters long in which friction and a changed discharge coefficient both matter; the stem is cut off so the hole stays the sharp-edged orifice Torricelli's law describes. The cut leaves a flat end; with r0 = 0.5 cm the missing point is h0 = 1.0 cm below the hole, and with 5 cm and 3 cm of water above the hole the outflow is 4.20 and 3.25 mL/s while the depth falls at 0.148 and 0.259 cm/s. C is the least certain input and is fitted from the learner's first readings. The learner knows it worked when the gradients read from the graph grow as the water gets shallower even though the outflow shrinks.
What changes
- What you change
- time since release
- What you measure
- water depth h
- What you keep the same
- same funnel and hole
- starting depth 8 cm above the hole
- water at room temperature
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- The level drops at a steady rate; it speeds up as the surface narrows even though the outflow slows.
- dh/dt and dV/dt are the same rate in different units; they are linked by the surface area, which changes with h.
Safety card
Hazards
- water on the floor
- drilling the hole
Controls
- the teacher drills the hole before the lesson
- work over a tray
Note
No chemicals and no heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; ordinary classroom supervision.
Curriculum references
The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.
- Mathematics Extension 1 11–12 Syllabus (2024), Year 12 focus area Further applications of calculus; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22ME1-12-05
- Mathematics Advanced 11–12 Syllabus (2024), Year 12 focus area Applications of calculus; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MAV-12-06
- Australian Curriculum v9No Australian Curriculum v9 code is listed.
Sources
The pages the author read to write this activity.