Assignment 12 – Using spreadsheets
 
Problem # 6
 
Explore problems of maximization such as the lidless box formed by a 5 x 8 sheet of paper with a square removed from each corner.
 
As can be seen from the diagram below, a lidless box can be produced from a sheet of paper by cutting a square out of each corner and folding up the sides.  Since the height must be uniform, each square must be the same size, and for the 5 x 8 paper, the square must be less than 2.5 on each side.
 
 
The volume of the box will be x(8-2x)(5-2x).  This is a cubic equation, and will have a maximum and a minimum volume. We can investigate the different sizes and volumes produced using Excel, by entering a value for x and having Excel calculate the other 2 dimensions and the volume.  We can then increment the values of x and see how the volume changes.
 

 
 
From the chart above it can be seen that the volume is maximized when the box is 1 x 3 x 6.
 
This problem can also be solved using Graphing Calculator, as shown below:
 

 
 
Problem #7:  Using Excel, place 4 numbers in the first row as follows
A  B  C  D
 
For each successive row replace the entries by the absolute value of the difference of the entry just above and the entry just to the right in the previous row.  In the 4th position use the absolute value of the difference of the fourth and the first rows.
|A-B|  |B-C|  |C-D|  |D-A|
 
The process always leads to zeros in all 4 entries within a few rows.
 
The largest number of rows before a zero row that I achieved is shown below,:  11 rows

So far I have only found this by guess and check methods, and have determined that at least some prime numbers to start with seems to help, and either increasing or decreasing consistently from A to D is also good. But, I believe that this number relationship is similar to what we call a step response in engineering, where after an initial impulse is input into a system, we look at the initial response until the response to the impulse dies down to what we call the steady state response. In an unstable system this can take a longer time, and in fact some systems never recover, and oscillate. Sometimes in engineering, and here, that is the reponse we want. So, I think maybe that it would be possible to come up with an unstable sequence of numbers, such that the steady state - all zeros, is never reached.