Maths IA · Calculus · topic 33 of 97
The most efficient wine glass: least glass for the volume · IB Maths IA idea: guide, sample graph and step-by-step plan
same drink, less glass?Every glass holds the same drink, but they do not all use the same amount of glass. You are going to look for the bowl shape that needs the least glass for a given volume, and see how far real glasses are from it.
- ✓ you like a problem where the shape itself is the unknown
- ✓ you want to combine volumes of revolution with an optimisation
- ✓ you would enjoy testing a result on a glass you can hold
The big idea
A bowl is a curve rotated about a vertical axis. Give the curve one shape parameter and you have a whole family of bowls, from a cone to a nearly straight-sided cup. Volume comes from slicing into discs, which is course calculus at HL. The glass needed comes from slicing into thin bands, and that formula is beyond the course, so you must explain where it comes from. Fix the volume, then ask which member of the family needs least glass. The optimum is rarely the shape of a glass on a table. The marks come from explaining the gap: what the real glass is doing that your model leaves out, such as stability, the stem, the rim and how it feels to drink from.
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 →
Use a power curve, then try a circular arc or a polynomial as the bowl profile, each with one or two shape parameters.
→ You compare families by the least area each can reach at the same volume, and say why one is flexible enough and another is not.Fix the height first and optimise the shape. Then let the height vary as well.
→ One variable becomes two. You decide what to hold fixed, and the answer depends on that choice.Include the flat base and the thickness of the glass, so that you count the glass as well as the surface.
→ The cost function gets extra terms, and the best shape moves. You can explain which term pulls it which way.Trace the outline of a glass on a grid, or photograph it, then fit your family to the points.
→ Your model meets data. You compare predicted and poured volume, and report the surface area of the real glass against the optimum.Use the trapezium rule or a spreadsheet for the area integral, and check it on a cone or a hemisphere with a known area.
→ You test your numerical method against an exact answer before trusting it on the family.Give the rim, the walls and the base different costs per square centimetre.
→ The cost function now has weights, as in a manufacturer's problem. Describe how the best shape changes with the weights.Your Maths IA research question: run the check
what is held fixed?Write your own question first. Then make sure it ticks every box:
- Names the family of curves and the shape parameter, and says why that family is reasonable for a bowl.
- Says what is held fixed (volume, height or both) and what is minimised.
- Says how the surface area will be calculated and how that calculation will be checked against a known shape.
- Names the real glass or glasses you will measure and what you will compare with the model.
- Includes a decision of your own (which family, which fixed quantity) and does not just apply one formula.
The model
A bowl is a profile curve turned round the vertical axis. Slice it into thin discs for the volume and thin bands for the surface. Draw your own glass and mark what you measured.
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.
Why this object, what 'efficient' means here, and what you expect before calculating: will the best bowl be tall or wide?
The profile, every symbol defined, and the volume as an integral you evaluate exactly.
Where the band formula comes from, a check on a cone or a hemisphere, and the integral for your family.
The constraint used to remove a variable, and the area as a function of the shape parameter.
The minimum found by calculus or by a numerical search, its meaning, and how it changes with the height.
Your measured glass beside the optimum: a number for the gap and a reason for it.
Back to your aim: the best shape, how far real glasses are from it, and why.
The family you chose, what the model ignores (stem, base, strength, taste), and which assumption matters most.
Where every formula and measurement came from.
What your Maths IA graph could look like
mark the minimum!Made-up numbers to show the form of a strong graph, not the answer: glass area against a shape parameter at a fixed volume. Yours will look different. example only
- Surface area against the shape parameter at one fixed volume, so the minimum is easy to see.
- Axis labels give units: area in square centimetres, and the parameter has none.
- Mark the minimum and say how it was found, not only where it is.
- Plot a second curve for another height to show the best shape depends on what you hold fixed.
- Another graph to consider: the measured profile of your real glass with the best-fitting member of your family.
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.
How much glass does each shape need?
Rank bowls of the same volume on one number: the surface area of the glass. Work it out for your own glasses and explain why the gaps are not equal.
Your game plan
Tick steps off as you go. We'll remember on this device.
0 of 10 done
- Read and predictLook at three glasses. Before any calculation, guess whether the shape that needs least glass for its volume is tall and thin or short and wide.
- Choose a familyPick a profile curve with one shape parameter that can produce both a cone-like and a bulging bowl. Say why it suits a glass.
- Find the volumeRotate the profile and integrate. Define every symbol, and check the formula on a shape with a known volume.
- Trial run (1 lesson)Measure a real glass at 8 to 10 heights and pour its volume. Is the glass really round? Where does the bowl start? Fix your method now.
- Learn the area formulaFind out where the surface-area formula comes from, draw the thin band, and test it on a cone or hemisphere. You must explain this yourself.
- Fix the volume and minimiseUse the volume to remove one variable, write the area as a function of the shape parameter and find its minimum. Check it with a graph.
- Compare with the real glassFit your family to your measured profile, then compare predicted and poured volume, and the glass's area with the optimum.
- Go further, with a reasonLet the height vary, add the base, or weight the costs. Say what question the extra step answers. HL: this is where the rigour for criterion E comes from.
- Reflect as you goAfter every result, write one line on what it means for a glassmaker and what it assumes. These lines become your evaluation.
- 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
- Volume of revolution – Rotate the profile about the vertical axis and evaluate exactly for the family.
- Surface area of revolution – Use thin bands, , to build the area integral, then test it on a cone.
- Constraint and elimination – Use fixed volume to write in terms of and , leaving one variable to optimise.
- Optimisation – Find with numerically, and use the second derivative or a graph to confirm a minimum.
- Numerical integration – Evaluate the area integral by the trapezium rule or software, with the step size reduced until the answer settles.
- Measurement and fitting – Fit the profile of a real glass and compare predicted and poured volume.
Key equations
Syllabus links
- 3.13D distance, volume and surface area
- 5.3Differentiating powers of x
- 5.8Maxima, minima, inflexion and optimisation
- 5.11Definite integrals and areas between curves
- 5.17Areas with the y-axis; volumes of revolutionHL
- 3.13D distance, volume and surface area
- 5.7Optimisation in context
- 5.8Trapezoidal rule
- 5.12Areas with the y-axis; volumes of revolutionHL
Read up first
Data and sources to start from
Trace or photograph the outline of one or more real glasses against a grid, and measure the poured volume with a measuring jug.
References to cite
- Stewart, J. (2015). Calculus: Early transcendentals (8th ed.). Cengage Learning. (Section 8.2, arc length and surface area of revolution.)
- Courant, R., & Robbins, H. (1996). What is mathematics? An elementary approach to ideas and methods (2nd ed., revised by I. Stewart). Oxford University Press.
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
test the area formula on a cone- Using the volume formula for the surface area. The area needs the slant of the curve, not just its radius.
- Forgetting to say what is counted: the outside only, the inside as well, the base, the rim.
- Optimising without fixing the volume. A smaller glass always needs less glass.
- Trusting a numerical area without testing the method on a shape whose area you know.
- Choosing a family that cannot produce a real glass shape, so the optimum says nothing about real glasses.
- Presenting the minimum as 'the best glass' without discussing stability, the stem and how it is used.
More Maths IA ideas like this
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Honeycomb: why do bees use hexagons?●○○
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.