Where should the 125 ml line go on a glass?

AA HLAI HL Solid Also in: Modelling

Research question

Using a volume of revolution of a fitted profile, at what height should the 125 ml and 175 ml lines be marked on a particular glass, and how accurate is the prediction?

Adapt it: change the place, the data or the comparison until the question is yours.

Why it makes a good exploration

Measures in bars and restaurants must be accurate by law in many countries. Modelling a real glass and predicting the height of a fill line gives an answer you can test with a measuring jug.

The mathematics you'll need

Fitting a polynomial or other function to the glass profile
Volume of revolution about the y-axis
Solving V(h) = 125 for h numerically
Error analysis

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

Photograph a stemmed glass with a ruler, extract profile points; test fill lines with a measuring syringe or jug.

GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.
Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Fit the inside profile x = f(y).
Set up V(h) = π∫₀ʰ x² dy.
Solve for heights giving 125 ml and 175 ml.
Test by filling and measuring; compare.
Discuss glass thickness and meniscus.

Pitfalls that cost marks

Modelling the outside rather than the inside of the glass.
Rotating about the wrong axis.
Forgetting the curved base.

Showing personal engagement

Choose a glass from home with personal significance.
Compare two glass shapes and discuss which makes pouring errors more obvious.
Link to consumer law on serving sizes.

See Criterion C: personal engagement for what examiners look for.

Taking it further

Find the rate at which the liquid level rises when poured at a constant rate (related rates) and explain why it is slow at the bowl's widest point.

Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Desmos workspacePlot your data, fit models and keep your graphs with your plan. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, a practising IA examiner, reads your draft and talks it through with you (€100).