What it shows
Cutting equal squares from the corners of a sheet and folding it gives a box. The volume rises and then falls as the cut grows, and calculus predicts exactly where it is greatest.
The science
If squares of side x are cut from the corners of a square sheet of side 24 cm, the box has base (24 minus 2x) by (24 minus 2x) and height x, so V = x (24 minus 2x) squared. Differentiating gives dV/dx = (24 minus 2x)(24 minus 6x), which is zero when x = 4 or x = 12. Only x = 4 is sensible, and the second derivative is negative there, so it is a maximum. Then V = 4 x 16 x 16 = 1024 cubic cm. The graph of V against x is a smooth hill, and the box is largest at the top of the hill.
Materials
Cost: under Rs 50. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- 6 sheets of square card, 24 cm by 24 cm
- Scissors and a ruler
- Tape
- Rice or sand to measure volume
- A measuring cup or a 250 ml container
- Graph paper
- A calculator
Safety
For this project: An adult must supervise the use of scissors. Wash hands after handling the rice.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
All the safety rulesStep by step
8 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
Write down the formula V = x (24 minus 2x) squared and make a table of x from 1 to 11 in steps of 1 with the calculated volumes.
Make real boxes by cutting squares of side 1, 2, 3, 4, 5 and 6 cm from the corners of six sheets and folding the sides up.
Tape the sides and fill each box with rice level with its top. Pour the rice into the measuring cup to read the volume.
Compare the measured volumes with the calculated ones. Differences come from taped corners and rounding of the rice.
Plot volume against x from the table, and draw a smooth curve through the points.
Differentiate V by expanding it: V = 4x cubed minus 96 x squared + 576 x, so dV/dx = 12 x squared minus 192x + 576, then solve for x = 4.
Show with the second derivative, d2V/dx2 = 24x minus 192, that it is negative at x = 4, so V is a maximum.
Repeat the problem for a rectangular sheet of 30 cm by 20 cm and predict the best cut using calculus before making the boxes.
What you should see
The real box with a 4 cm cut holds the most rice, close to 1,024 cubic cm, and the calculus answer matches the peak of the experimental graph.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
1What does dV/dx = 0 tell us?
It gives the points where the volume stops rising and starts falling, which are maximum or minimum candidates.
2Why do we reject x = 12?
A cut of 12 cm leaves no base, so the volume is zero.
3How can you tell the answer is a maximum?
The second derivative is negative there, or the volumes on both sides are smaller.
4Where is this idea used in industry?
Designing packaging and containers to hold the most for the least material.
Take it further
Find the shape of a cylinder-shaped tin with the least metal for a fixed volume of 500 ml and compare it with a real tin.
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
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