The box that holds the most
You cut the same size square from each corner of a 30 cm × 20 cm sheet of cardboard. Then you fold up the flaps, and the sheet becomes a box with no lid.
What size cut makes the box that holds the most?
Explore
Make your guess first, in task 1. The Cut unlocks after that.
Cuts table
Measurements in cm; volume in cm³.
| Cut | Length | Width | Height | Volume |
|---|
Find the box that holds the most
Before you move the Cut, lock in what you expect.
Then move the Cut and watch the height and the bottom of the box. When you find the biggest volume, test its neighbors too: 0.5 cm less and 0.5 cm more.
Now read What's going on, right below, and then tap Next.
A square sheet
Change the Sheet to 30 cm × 30 cm. Before you move the Cut, pick the cut you think works best for this sheet.
Find the cut that makes the box that holds the most, and test its neighbors. Is the best cut the same as on the other sheet? What about the volume?
Two boxes of almost 1,000 cm³
Set the Sheet back to 30 cm × 20 cm. Find two different cuts that make boxes between 990 cm³ and 1,010 cm³.
More than 1,056 cm³
On the 30 cm × 20 cm sheet, turn on Fine step and find a cut with a volume greater than 1,056 cm³.
With a cut of centimeters on this sheet, the volume is
Why does have to stay below 10?
What's going on
Each side of the sheet loses a square at both ends. So with a 3 cm cut, the length of the bottom is 30 − 3 − 3 = 24 cm, and the width is 20 − 3 − 3 = 14 cm. The height of the box is the cut itself: 3 cm.
The volume of a box is how many small cubes, 1 cm on each side (1 cm³), fit inside it. For a rectangular box, you get it by multiplying its three measurements:
The last paragraph shows up when you finish task 1.
A small cut gives a big bottom, but the box is shallow. A big cut makes the box tall, but the bottom shrinks on both sides. The volume goes up as long as the height you gain makes up for the bottom you lose. It reaches a peak and then goes down. On the graph, the points make a hill.
For teachers
Objective
By the end, students can find the cut that makes the box with the greatest volume from a sheet cut at the corners, calculating the volume of each cut they test and explaining why cutting too little or too much gives less. It applies the volume of a right prism (area of the base times height) from section 10.7 to an optimization problem solved without derivatives, by testing.
Prerequisites
Multiplying and subtracting decimals, the idea of volume as a number of 1 cm³ cubes, and reading points on a coordinate plane. The expression for the volume in terms of appears only in Going further, the last task in the panel, for students who already work with algebraic expressions.
How to use in class
About 25 minutes. If you can, start with a real sheet of paper or cardboard and a pair of scissors: cut a square from each corner in front of the class and fold it (2 min). Project the activity and have each student, or each pair, write their guess on paper before locking it in: whether the volume only goes up, only goes down, or goes up and then down, and which cut gives the most (3 min). Let the class move the Cut for 7 minutes, with a question on the board: "what happens to the height and to the bottom when the cut gets bigger?" Stop and collect the biggest volumes they found; ask someone to read the 4 cm calculation out loud (22 × 12 × 4). Read "What's going on" and the check question together (5 min). Wrap up with tasks 2 and 3 in the panel, the square sheet and the two cuts near 1,000 cm³ (8 min).
Common mistakes
Many students guess a small cut ("there's more cardboard left") or a big cut ("it gets taller"). The locked-in guess makes this visible: compare the class's guesses with the graph. Another mistake is taking the cut off one end only and saying the length is 30 − 4 = 26 cm; the check question gives feedback for this case (answer 14 instead of 8), and the flat sheet shows the two squares on each side. Some students also think the same sheet always gives the same volume, because the cardboard is the same; the numbers in this activity go from 275.5 cm³ to 1,056 cm³ with the same sheet. Finally, some conclude that 4 cm is always the best cut. On the 30 cm × 30 cm square sheet the best is 5 cm (2,000 cm³), and there the 4 cm cut gives 1,936 cm³; with another sheet, such as 40 cm × 30 cm, the best is close to 5.7 cm.
Discussion questions
- Why can't the cut reach 10 cm on the 30 cm × 20 cm sheet?
- The 3 cm and 5 cm cuts give 1,008 cm³ and 1,000 cm³. Which of the two boxes would you use to hold colored pencils? And to hold a notebook?
- On the 30 cm square sheet, the best cut is 5 cm, which is one sixth of the side. Try it on paper with a 12 cm square sheet: is the best cut still one sixth of the side?
- When the cut gets bigger, does the area of cardboard that becomes the box go up or down? And the volume?
- If you were selling boxes and cardboard were priced by the square centimeter, would you still pick the cut with the greatest volume?
Based on Contemporary Mathematics, OpenStax, section 10.7, under the CC BY-NC-SA 4.0 license. This activity is under the same license. Access the book for free at openstax.org.