Cones & spheres · Ages 14–15
Folding a box
V = x(length − 2x)(width − 2x); a table finds the greatest.
How to do it
Method
V = x(length − 2x)(width − 2x); a table finds the greatest
20 × 16, cut 3: 3 × 14 × 10 = 420 cm³
Another way
Draw V against x
Read off the curve's highest point
In full
Cut a square of side x from each corner of a rectangular sheet and fold up the sides, and you have a box with no lid. It is x high, its base (length − 2x) × (width − 2x), so V = x(length − 2x)(width − 2x). A 20 cm × 16 cm sheet with 3 cm cut out: 3 × 14 × 10 = 420 cm³. Too small an x makes a flat box, too big a tiny base: somewhere between, an x makes the volume greatest. Try them in a table, x = 1, 2, 3, ...: the volume rises and then falls, and near the top you can try decimals. x can't be half the width or more, or there is no base. A sheet of A4 is 29.7 cm by 21 cm: cutting out 4 cm gives 4 × 21.7 × 13 = 1128.4 cm³, more than 3 cm or 5 cm gives. Another way: draw V against x and read off the highest point.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
A 21 cm × 13 cm sheet, as in the picture, has squares of side 1 cm cut from its corners and is folded into a box with no lid. The base is cm long and cm wide, and the box cm high.
Fill in each blank
Show the answer and working
Answer: 19, 11, 1
Each side loses 1 at both ends: 21 − 2 × 1 = 19, 13 − 2 × 1 = 11; the sides folded up are 1 high.
Example 2
A square sheet of side 25 cm has squares of side 4 cm cut from its corners and is folded into a box with no lid. How long is a side of the box's base?
- A. 9 cm
- B. 21 cm
- C. 25 cm
- D. 17 cm
Choose the right answer
Show the answer and working
Answer: D. 17 cm
25 − 2 × 4 = 17 cm.
Example 3
A square sheet of side 24 cm, as in the picture, has squares of side x cm cut from its corners (x a whole number) and is folded into a box with no lid. What x makes the box hold the most?
Show the answer and working
Answer: 4 cm
Tabulate V = x(24 − 2x)2: it rises and then falls, greatest at x = 4 (1024 cm³) — a sixth of the side.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.A 25 cm × 22 cm sheet, as in the picture, has squares of side 1 cm cut from its corners and is folded into a box with no lid. The base is cm long and cm wide, and the box cm high.
Fill in each blank
Answer:
2.A 21 cm × 19 cm sheet, as in the picture, has a square of side 4 cm cut from each corner and is folded into a box with no lid. What is the box's volume, in cm³?
Answer:
3.A square sheet of side 18 cm, as in the picture, has squares of side 3 cm cut from its corners and is folded into a box with no lid. How much does the box hold, in cm³?
Answer:
4.From a 25 cm × 16 cm sheet, the box made by cutting 3 cm squares from the corners holds more than the box made by cutting 2 cm squares.
True or false?
Answer:
5.A square sheet of side 36 cm, as in the picture, has squares of side x cm cut from its corners (x a whole number) and is folded into a box with no lid. What x makes the box hold the most?
Answer:
All the answers
- 1. 23, 20, 1
- 2. 572 cm³
- 3. 432 cm³
- 4. True
- 5. 6 cm
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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