Mathematics 11–12 · Years 11–12

The open box: cut squares from a sheet and find the largest volume

Applications of calculus: Optimisation (Mathematics Advanced, Year 12); Working with functions: Quadratic and cubic functions (Mathematics Advanced, Year 11); Polynomials: Remainder and factor theorems (Mathematics Extension 1, Year 11); Polynomials: Sums and products of zeroes of polynomials (Mathematics Extension 1, Year 11)

Practical, model not builtLow risk

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The idea

A cubic volume function built from a real sheet has a maximum that calculus locates and a measured fill of rice confirms.

What you need

Six squares of card 20 cm by 20 cm (the NRICH Cuboid Challenge size) and two A4 sheets (297 mm by 210 mm)
Ruler, scissors, sticky tape
Dry rice and a 2 L measuring jug (or a 1 L jug filled twice for the A4 box, which holds 1.128 L)

How to do it

From the 20 cm squares cut corner squares of side 2, 3, 3.3, 4 and 5 cm, then fold and tape each box.
Fill each box level with rice, pour the rice into the jug and read the volume; tabulate x against measured volume.
Write V(x) = x (20 - 2x)^2 for 0 < x < 10, differentiate, solve V'(x) = 0 and classify the stationary points with the second derivative.
Repeat for A4 with V(x) = x (297 - 2x)(210 - 2x) in millimetres; solve the quadratic V'(x) = 0 and build the box at the optimum.
Compare the measured and calculated volumes and explain why the measurement sits below the formula.
Find every cut that gives a box of exactly 576 mL: show that x = 4 is a zero of 4x^3 - 80x^2 + 400x - 576 by the factor theorem, divide to find the other zeroes, check that they sum to 20, and build and fill both boxes that fit the domain.

What you should see

For the 20 cm square, V'(x) = 0 at x = 10/3 = 3.33 cm with V = 592.6 cm^3; the cuts made give 512.0 (x = 2), 588.0 (x = 3), 592.5 (x = 3.3), 576.0 (x = 4) and 500.0 cm^3 (x = 5), so the curve is flat near its peak: the volumes for x = 3, 3.3 and 4 cm lie within 16.5 mL of each other (under 3 per cent), inside the reading and levelling error of a kitchen measuring jug, while those for x = 2 and 5 cm sit at least 64 mL lower. The other root, x = 10, is an end of the domain where the volume is 0. For A4, V'(x) = 12x^2 - 2028x + 62370 = 0 gives x = 40.4 mm and V = 1.128 L. A 576 mL box comes from x = 4 cm or x = 8 - 2 sqrt 7 = 2.71 cm; the third zero, 8 + 2 sqrt 7 = 13.29 cm, lies outside the domain, and the three zeroes sum to 20 and multiply to 144 as the coefficients predict. Measured volumes sit below the formula because card thickness and rounded folds reduce the inside dimensions. The learner knows it worked when the measured volumes for x = 2 and 5 cm are clearly below those for 3, 3.3 and 4 cm, which the jug cannot separate, and the calculus then places the maximum at 3.33 cm.

What changes

What you change: side length x of the cut-out square
What you measure: volume of the folded box
What you keep the same: sheet size
fill material and levelling method
same jug

Common misconceptions

Each of these ideas is wrong, and the activity is a chance to test it.

The biggest box comes from the biggest cut; large cuts remove base area faster than they add height.
Every solution of V'(x) = 0 is an answer; for the square sheet the root x = 10 cm gives a box with no base.

Safety card

Low riskLearners carry it out

Hazards

scissors

Controls

cut on a table with the blades pointing away from the body

Note

No chemicals and no heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; ordinary classroom supervision.

Curriculum references

The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.

Mathematics Advanced 11–12 Syllabus (2024), Year 12 focus area Applications of calculus; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MAV-12-06
Mathematics Advanced 11–12 Syllabus (2024), Year 11 focus area Working with functions; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MAV-11-02
Mathematics Extension 1 11–12 Syllabus (2024), Year 11 focus area Polynomials; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22ME1-11-02
Australian Curriculum v9No Australian Curriculum v9 code is listed.

Sources

The pages the author read to write this activity.

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