Maths IA · Calculus · topic 9 of 40
Cans: the least metal for a given volume · IB Maths IA idea: guide, sample graph and step-by-step plan
gentle start, deep finish: the gap is the IADifferentiate once and the textbook 'best can' falls out in a line. Your IA starts where that line stops matching the cans on the shelf.
- ✓ you like a clean problem built on a real object
- ✓ you can bring cans in from the kitchen
- ✓ you want calculus you can fully explain
The big idea
For a fixed volume, a cylinder's surface area depends on one variable, so calculus can find the shape that uses least metal. That part is routine, and on its own it will not earn much. The real interest is the gap: real cans are taller and thinner than the ideal, so something the simple model ignores must matter. Seams, thicker lids, the rim, stacking, how a can feels in the hand. Build a better model, say what each change does to the best shape, and test it on cans you have measured. Predict first, then check.
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 →
Give the lid and base a different thickness from the side wall, then minimise the cost of the metal.
→ Your function now has an extra parameter. The best shape moves, and you can describe how it moves as the thickness ratio changes.Include the side seam and the double-seamed rims, whose length depends on the circumference.
→ New terms in the function that are not simply proportional to area, so the stationary point is no longer the textbook one.Measure 10 or more cans of different sizes and compare their proportions with your model's best shape.
→ Your graph becomes model against data. You need a measure of how far each can is from the ideal, and a reason for any pattern.A square-based box, a hexagonal prism or a rounded carton, with the same volume constraint.
→ A new function to build and optimise, and a comparison between shapes that has to be fair (what is held equal?).Count the wasted space on a shelf or in a box for round and square containers.
→ The quantity you minimise is now metal plus packaging per unit volume, which is a better question for a real company.Keep V as a symbol throughout and see how the best radius scales with it.
→ Your result becomes a general statement you can justify algebraically, not a single number for 330 mL. This is where HL-level rigour shows.Your Maths IA research question: run the check
what is held fixed? say it!Write your own question first. Then make sure it ticks every box:
- Says what is minimised (metal area, metal cost, or total packaging) and what is held fixed (volume).
- Says which real-world features are in the model (lids, seams, rims, wall thickness) and which are left out on purpose.
- Says what the model will be compared with: measured cans, a published design, or another shape.
- Needs something beyond the standard one-variable result, so the work is yours and not a textbook exercise.
- Is small enough to finish in 12 to 20 pages, with maths from the course that you can explain line by line.
The model
A cylinder and its net: the pieces whose areas you add up. Draw yours with your own symbols and add the extra pieces your model includes.
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 cans, which cans you chose, and what 'best' will mean (least metal? least cost?). One sentence on what you expect before you calculate.
Every symbol defined. Surface area written as a function of one variable, with the volume constraint used to remove the other.
Derivative, stationary point, and a reason it is a minimum (second derivative or sign change). Do it with V kept as a symbol if you can.
Which cans, which tool, how you measured, a labelled table with units and uncertainties in the headers.
Each extra feature added one at a time, with a reason, a new function, and what it does to the best shape. A graph comparing versions.
A graph of ideal, improved model and real cans on the same axes. A number for how close each is, not 'quite close'.
Back to your aim. Does the best shape move, by how much, and why?
Which assumption hurts most? Limits of measuring by hand, of ignoring manufacturing, of the cylinder itself. At least one point made earlier in the body too.
Every source for measurements, designs and theory.
What your Maths IA graph could look like
say how FLAT the minimum isMade-up values, to show the form of a strong graph, not the answer. Yours will look different. example only
- Axis titles say what is plotted and give units, with the uncertainty of the quantity you chose.
- Model and data share the axes, so the reader sees the gap without hunting for it.
- The scale starts near the data, not at zero, so the shape of the minimum is visible. Say so in the text.
- Mark the minimum and say how flat it is: a flat minimum matters when you compare cans.
- Another graph to consider: percentage extra metal against the height-to-diameter ratio, for all your cans.
Your game plan
Tick steps off as you go. We'll remember on this device.
0 of 10 done
- Read and predictFind a source for how cans are made. Before any calculus, write down what you think the best shape looks like and which real feature you expect to push cans away from it.
- Build the ideal modelDefine symbols, write the area as one function, and decide which variable to eliminate. Check the units.
- Optimise it properlyFind the stationary point, show it is a minimum, and try it with V as a symbol so the result is general.
- Trial run (1 lesson)Measure three cans. Where do you measure the diameter? Which height counts, with or without the rim? Fix your definitions now.
- Collect the dataMeasure enough cans, of different sizes, with a consistent method. Record the volume printed on each one.
- Pick one improvementChoose the real feature you think matters most. Add it to the model and see whether the best shape moves the way you predicted.
- Compare with the cansPlot model and data together. Find a way to measure the distance between them and decide what a 'close' fit would be.
- Go further, with a reasonAdd a second feature, or generalise the algebra, or compare with a box. HL: this is the step that builds the sophistication and rigour for criterion E.
- Reflect as you goAfter every result, write one line on what it means and what it assumes. Later, these become your evaluation.
- Cut, check, then sendCheck every graph has titled axes with units, every symbol is defined once, and the conclusion answers your aim. Then give your teacher one draft.
Maths you'll need
Traps to avoid in this Maths IA
HL needs more than this!- Stopping at the textbook result: the one-variable optimisation is SL work, and at HL it is routine unless you push it further.
- Forgetting to show the stationary point is a minimum, not just a stationary point.
- Using the volume printed on the can, not the volume of the container you measured, without saying which.
- A model with seams and thicknesses but numbers pulled from nowhere. Measure them or cite them.
- Measuring the diameter once with a ruler and quoting it to 0.01 cm.
- Concluding that 'real cans are not optimal' without a number for how far off they are.
- Calculator notation such as r^2 in the written working instead of proper notation.
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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.