IA idea · Calculus & optimisation
Where is the best seat in the cinema?
Research question
In which row of my local cinema is the vertical viewing angle of the screen greatest, and how does raked seating change the answer?
Make it yours, then agree it with your teacher.
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Why it makes a good exploration
Sit too close and you look up at the bottom of the screen; too far and the screen shrinks. The angle the screen subtends has a maximum you can find with calculus — the same problem the astronomer Regiomontanus posed in 1471.
The mathematics you'll need
- Inverse trigonometric functions to express the viewing angle
- Differentiation (or technology) to maximise the angle
- Coordinate geometry for raked (sloped) seating
- Comparing with industry viewing-angle guidelines
Where the data comes from
Estimate screen height, screen bottom height and row spacing (pace it out, or use published seat plans).
- Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.
- GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.
A possible outline
- Set up θ(x) = arctan((a + h)/x) − arctan(a/x).
- Find the maximum analytically or with technology.
- Add raked seating: eye height rises with each row.
- Compare with recommended viewing angles.
- Reflect: horizontal angle, comfort, sound.
Pitfalls that cost marks
- Measuring from the floor instead of eye level.
- Not justifying the assumption that a bigger angle is better.
- Leaving the derivative unsimplified and unverified.
Showing personal engagement
- Measure your own favourite cinema or school hall.
- Compare IMAX and standard screens.
- Survey friends' preferred rows and compare with your model.
See Criterion C: personal engagement for what examiners look for.
Which course is it for?
| Course | Fit | Maths to lean on |
|---|---|---|
| AA SL | Good fit | Inverse trigonometric functions to express the viewing angle; Differentiation (or technology) to maximise the angle |
| AA HL | Good fit | Inverse trigonometric functions to express the viewing angle; Differentiation (or technology) to maximise the angle |
| AI SL | Good fit | Inverse trigonometric functions to express the viewing angle; Differentiation (or technology) to maximise the angle |
| AI HL | Good fit | Inverse trigonometric functions to express the viewing angle; Differentiation (or technology) to maximise the angle |
Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.
How this idea reaches the top bands
Reflection (D)
For this idea, start with the weak points: measuring from the floor instead of eye level; and not justifying the assumption that a bigger angle is better. Say how each one affects your answer.
Use of mathematics (E)
SL: inverse trigonometric functions to express the viewing angle; differentiation (or technology) to maximise the angle; coordinate geometry for raked (sloped) seating; comparing with industry viewing-angle guidelines, each explained in your own words and carried out correctly.
How calculus explorations reach the top bands in general: Calculus & optimisation IA ideas. Guides: A · B · C · D · E · self-grade sheet.
Taking it further
Optimise both vertical and horizontal viewing angles together, or find the seat that minimises neck strain.
Extending it for HL
Add an HL technique the question genuinely needs: what an HL calculus exploration adds.
See a complete IA, marked
Is a hanging chain a parabola? Comparing catenary and quadratic models (AA SL) is a complete calculus exploration, marked on every criterion.
Frequently asked questions
Which IB Maths course is this IA idea for?
It works for AA SL, AA HL, AI SL and AI HL. We rate it Solid: it needs some independent work beyond class examples.
What mathematics does this IA use?
Inverse trigonometric functions to express the viewing angle, differentiation (or technology) to maximise the angle, coordinate geometry for raked (sloped) seating and comparing with industry viewing-angle guidelines.
Where can I get data for this IA?
Estimate screen height, screen bottom height and row spacing (pace it out, or use published seat plans). Checked sources and tools: Desmos graphing calculator and GeoGebra.
Turn this idea into your IA
The planner opens with “Where is the best seat in the cinema?” filled in: the question, the maths and the data sources.
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