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Maths IA · Calculus · topic 22 of 97

Where should a screen be mounted? The best viewing angle · IB Maths IA idea: guide, sample graph and step-by-step plan

closer is not better

Sit too close to a screen and it fills your view badly, sit too far and it shrinks. Somewhere between, there is a best seat, and you can find it exactly.

Difficulty●●○ Medium
KitA tape measure, a room with a screen or board, a phone clinometer app or a protractor, a spreadsheet
TimeAbout 12 to 15 hours
FitsAA SL3 of 3AA HL3 of 3AI SL2 of 3AI HL3 of 3
  • ✓ you like a problem you can measure in a room you already know
  • ✓ you want to see calculus choose a best seat
  • ✓ you are curious how a famous medieval problem still applies to screens

The big idea

The angle a screen fills in your view depends on how far away you sit. Treat that angle as a function of distance, using the difference of two inverse tangents, and it rises, peaks and falls. Differentiate to find the peak. The method goes back to Regiomontanus in 1471, who asked it about a painting hung above eye level. The best distance turns out to depend on the two edge heights in a surprisingly simple way. A good IA does not stop at one viewer. It asks what a whole room of viewers needs, what neck strain and picture quality add to the choice, and how real measurements compare with the model.

Make it yours: pick one twist

Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.

Swipe for more twists →

Many viewers

Model a seating area and find the best mounting height for all the seats together, for example by averaging the angle over the seats.

→ One optimum becomes a compromise. You decide how to combine viewers, and explain why that choice matters.
Add neck strain

Add a second function for how far a viewer tilts their head, and find the best distance for a weighted sum of the two.

→ A cost function with weights you must justify, and an optimum that moves back from the pure viewing-angle answer.
Tilt and height together

Let the mounting height and the tilt of the screen both vary, and study the viewing quality as a function of two variables.

→ Partial derivatives enter. You find the best pair, and show how flat or sharp the best region is.
Measure a real room

Measure the screen edges and eye heights in a classroom or cinema, then measure the angle at several seats with a phone clinometer.

→ Your model meets data. You compare predicted and measured angles, and explain the differences.
Eye level inside the screen

Look at the case where the viewer's eye is level with part of the screen, so one edge is below eye level.

→ The sign of one edge changes and the formula for the best seat no longer applies. You find out what happens instead.
Whole-seating integral

Replace the sum over seats by an integral over the area, with the viewing angle as the integrand.

→ A double integral appears. You compare it with the discrete sum and decide whether the extra step is worth it.

Your Maths IA research question: run the check

what does 'best' mean here?

Write your own question first. Then make sure it ticks every box:

  • Names the screen, its edge heights above eye level and the seating area, with measurements.
  • Says exactly what is being optimised: the viewing angle for one viewer, or a combination for many.
  • Says how the model will be tested against measurements in a real room.
  • States what is left out: screen brightness, glare, resolution, how far the viewer's head moves.
  • Is narrow enough for 12 to 20 pages and includes a decision of your own, such as how to weigh comfort against angle.
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The model

A screen on a wall, two heights above eye level, and a viewer at a distance. Draw your own room and mark every measurement you will need.

eye levelviewer's eyeabxθscreenθ(x) = arctan(b/x) − arctan(a/x)a, b: edges above eye levelthe angle shrinks ifyou are too close or too farwhere is θ greatest?draw your room: eye height and screen edges

How your Maths IA could be laid out

The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.

Aim and context~1 page
Why this room or screen, what 'best' means, and what you expect before calculating: closer or further than where people sit now?
The viewing angle~1.5 pages
Every symbol defined, the angle as a difference of two inverse tangents, and its graph for your screen.
The best distance~2 pages
The derivative, the equation for the maximum, its solution and a check that it is a maximum.
Comfort and many viewers~2.5 pages
The extra function or the sum over seats, how you weighted it, and the new best choice.
Model against real measurements~2 pages
Angles measured at several seats beside the predicted ones, with uncertainty and a reason for each gap.
What the answer changes~1.5 pages
How the best distance moves when the screen is raised, lowered or made larger.
Conclusion~1 page
Back to your aim: the best distance or height for your room, and how sure you are.
Evaluation~2 pages, the biggest part!
Which parts of 'a good view' your model ignores, how real seats are fixed, and which assumption matters most.
Bibliographynot counted
Where every formula, measurement and standard came from.

What your Maths IA graph could look like

mark the peak!

Made-up numbers to show the form of a strong graph, not the answer: viewing angle against distance, with measured angles from a room. Yours will look different. example only

0246810120510152025Distance from the wall x / mViewing angle θ / degreesθ(x) from the modelyour measured angles (made-up)the peak is broad: seats near itare almost as goodILLUSTRATIVE · MADE-UP DATA
  • Angle against distance, so the rise, the peak and the slow fall are all visible.
  • Axis labels give units: distance in metres and angle in degrees.
  • Mark the calculated peak, then show measured angles as points with error bars.
  • If your points sit below the curve, say what the model leaves out.
  • Another graph to consider: the best distance against the height of the bottom edge.

The maths in action

worked through, not sketched!

The graph above was a sketch with made-up numbers. This figure works the idea through from the model itself, so you can see what a finished figure looks like. The source line under it says where every number comes from.

Two panels. Left: a screen on a wall seen from three seats on the eye-level line, with a dotted circle through the screen edges touching the eye-level line at the best seat. Right: three humped curves of viewing angle against distance, with peaks moving right and getting lower as the screen is raised.
(a) A screen 2 m tall with its bottom edge 2 m above eye level, seen from three seats: the angle is 14 degrees at 1.2 m, 19 degrees at the best distance 2.83 m, and 14 degrees at 7 m. (b) The angle against distance for three mounting heights.Source: Computed from the model (no measured data).

How does the view change with the seat?

Rank five seats on one number: the angle the screen fills. Work it out for your own room and explain why the best seat is not the closest.

1.2 m14.3°2 m18.4°2.83 m (best)19.5°5 m16.9°7 m13.8°the closest seat is not the best: a seat at 1.2 m sees about the same angle as one at 7 mViewing angle in degrees, screen 2 m tall, bottom edge 2 m above eye levelILLUSTRATIVE · MADE-UP NUMBERS

Your game plan

Tick steps off as you go. We'll remember on this device.

0 of 10 done

  1. Read and predictFind out what the viewing angle problem is. Before calculating, guess whether the best seat is nearer the wall or further away than where you usually sit.
  2. Choose the roomPick a classroom, cinema or living room. Measure the screen edges and the eye height of a seated viewer, and say how you measured.
  3. Build the angleWrite the angle with inverse tangents, graph it for your screen and read an approximate peak before differentiating.
  4. Trial run (1 lesson)Measure the angle at two seats with a phone clinometer or a protractor and compare with the formula. Is the method good enough? Fix it now.
  5. DifferentiateFind the derivative of the arctan terms, set it to zero and solve for the distance. Show it is a maximum.
  6. Check the formulaTest the best-distance formula against your graph and against a table of values from a spreadsheet.
  7. Add something of your ownAdd several viewers, neck strain or the tilt of the screen. Say what question the extra step answers.
  8. Go further, with a reasonMove to two variables or to an integral over seats, and compare with the simpler model. HL: this is where the rigour for criterion E comes from.
  9. Reflect as you goAfter every result, write one line on what it means for a real room and what it assumes. These lines become your evaluation.
  10. Cut, check, then sendCheck units everywhere, every graph labelled, every symbol defined once and the conclusion answers your aim. Then send your teacher one draft.

The maths you'll use

  • Inverse trigonometric functions – Write the angle as , check it with right-angled triangles.
  • Differentiation – Differentiate each arctan term with the chain rule and set to get .
  • Optimisation – Show the stationary point is a maximum with changing sign or the second derivative, and describe how flat the peak is.
  • Sensitivity and graphs – Plot and against the edge heights and describe how they change.
  • Weighted cost function – Combine viewing angle with neck tilt in one function of , and find how the weights move the optimum.
  • Measurement and error – Measure heights and angles with uncertainty, and compare predicted and measured angles.

Key equations

Syllabus links

  • 3.2Right-angled trigonometry, sine and cosine rules
  • 5.3Differentiating powers of x
  • 5.6Standard derivatives; chain, product and quotient rules
  • 5.8Maxima, minima, inflexion and optimisation
  • 3.9Reciprocal and inverse trigonometric functionsHL
  • 5.15Further derivatives and integrals; partial fractionsHL

Read up first

Data and sources to start from

Measure the edge heights of a screen and the eye height of a seated viewer, then measure the viewing angle at several seats with a phone clinometer or a protractor.

References to cite

  1. Maor, E. (1998). Trigonometric delights. Princeton University Press.
  2. Stewart, J. (2015). Calculus: Early transcendentals (8th ed.). Cengage Learning.

Cite what you actually use, in your own format, and say where each number came from.

Traps to avoid in this Maths IA

heights from EYE level
  • Measuring heights from the floor, not from the viewer's eye level, so the formula gives the wrong seat.
  • Writing the angle as one arctan of the ratio of heights. It is the difference of two angles, not the angle of a difference.
  • Calling the maximum a minimum or never checking which it is.
  • Using one viewer's eye height for a room of people of different heights, and not saying so.
  • Reporting the best distance to three decimals when the eye height is only known to a few centimetres.
  • Forgetting that the formula assumes the viewer's eye is below the bottom edge of the screen.

More Maths IA ideas like this

Make sure it's yours

This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.

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