Under the ribbon
Problem
A ribbon is nailed down with a small amount of slack.
The distance between the nailed ends is 1 metre and the amount of ribbon is 104 cm.
What is the largest cube that can pass under the ribbon ?
And where along the ribbon is the best place for the cube to pass under, or doesn't the position matter ?
Getting Started
Start with a diagram of a cube just passing under a ribbon.
And maybe put the cube exactly in the middle, just to start with.
Next choose a size for the cube, any size, and calculate how long the ribbon would be if that was the biggest cube that would fit.
Was it 104 cm ? Adjust the cube size and try again ?
Student Solutions
Many people thought that the middle position would be best - that the cube size which only just fits under in the middle would not fit under anywhere else . . . . however no one has yet offered a reason for believing this . .A spreadsheet may help, though only if we can reason correctly about what the spreadsheet shows. You may like to look at a spreadsheet created for this problem.
Excel file : UnderRibbon
First you need to select a size of cube, the final column will then show the amount of ribbon needed for that cube placed at various positions along the metre length. However you will still need to reason quite carefully before you can be sure that the centre is the best position.Once you are happy about that, the Excel file has a second sheet called 'Cube in the Middle' (see the blue tab at the bottom of the page).This sheet represents a 'trial and improvement' method for a cube placed at the mid-point. You select the cube size to start at and also the size of the increase row on row, the final column will report the total length of ribbon required. That needs to be as close to 104 as you can manage without exceeding it. Although no one really accounted for their choice of the middle position, some good work was done with that position assumed.
Logan from McQueen High School, Colin from Northcote College, Keith from Nelson and Colne, and Sudheendra from Bangalore, all successfully arrived at a cube side length of 13.3178 by solving equations.
Here's how their algebra went :
x is used to represent the side length for the cube, and the space either side of the cube is expressed in terms of x.
The slope length of the ribbon can then be expressed in terms of x, by using Pythagoras, and hence the whole ribbon length for a cube of side x is :
This expression has been set equal to 104 so that we can try to solve the equation for x
Removing x from each side :
Removing the brackets and tidying up :
A negative solution might fit the equation but the expression is only being used to represent a context where x has a positive value.
Teachers' Resources
One reason for using this problem with students is to show how a spreadsheet approach can give access to problems where other analysis, in this case algebra, is very difficult.
A second reason is that the answer to the problem is surprising.
Somehow the brain doesn't expect such a large answer - perhaps we imagine that because there isn't much extra length extending the ribbon ( 4 cm of slack ) there will not be much scope for displacing the ribbon vertically.
This problem also needs approaching in stages, and it is good for students to experience that by solving for the symmetric central position first then giving consideration to the asymmetric case.
Perhaps students could extend the enquiry to consider how different degrees of slack : 3 cm, 2 cm, 1 cm, effect the result.
Some students will be very familiar with using a spreadsheet to help with this kind of mathematical context, but many will not be at all confident, or even think to try a spreadsheet.
Making calculations for several different cube sizes on a calculator first, helps students to see that the calculation process is the same each time only using different input values. It also helps students to grasp that the ability to cover input values at intervals systematically across a chosen range could guide us closer and closer towards an optimal position or value.