An open box is to be made from a sheet of card. Identical squares are cut off the four corners of the card, as shown below.
The card is then folded along the dotted lines to make the box.
My main aim is to determine the size of the square cut which makes the volume of the box as large as possible for any given square sheet of card.
I am going to begin by investigating a square with a side length of 24 cm. Using this side length, the maximum whole number I can cut off each corner is 11cm, as otherwise I would not be able to make an open cube.
I am going to begin by looking into whole numbers being cut out of the box corners.
The formula needed to get the volume of a box is:
Volume = Length x Width x Height
If I am to use a square of side length 24cm, then I can calculate the side lengths minus the cut out squares using the following equation.
Volume = Length - (2 x Cut Out) x Width - (2 x Cut Out) x Height (Cut Out)
Using a square, both the length & the width are equal. I am using a length and width of 24cm. I am going to call the cut out "x." Therefore the equation can be changed to:
When x = 1,
Volume = (Width - X) x (Length - X) x (Width -> also known as X)
My formula allows me to construct a spreadsheet, which would allow me to quickly and accurately calculate the volume of the box. Below are the results I got through this spreadsheet.
Here I have tabulated my results:
I have highlighted the value of X, which allows the maximum volume in each case
The card is then folded along the dotted lines to make the box.
My main aim is to determine the size of the square cut which makes the volume of the box as large as possible for any given square sheet of card.
I am going to begin by investigating a square with a side length of 24 cm. Using this side length, the maximum whole number I can cut off each corner is 11cm, as otherwise I would not be able to make an open cube.
I am going to begin by looking into whole numbers being cut out of the box corners.
The formula needed to get the volume of a box is:
Volume = Length x Width x Height
If I am to use a square of side length 24cm, then I can calculate the side lengths minus the cut out squares using the following equation.
Volume = Length - (2 x Cut Out) x Width - (2 x Cut Out) x Height (Cut Out)
Using a square, both the length & the width are equal. I am using a length and width of 24cm. I am going to call the cut out "x." Therefore the equation can be changed to:
When x = 1,
Volume = (Width - X) x (Length - X) x (Width -> also known as X)
My formula allows me to construct a spreadsheet, which would allow me to quickly and accurately calculate the volume of the box. Below are the results I got through this spreadsheet.
Here I have tabulated my results:
I have highlighted the value of X, which allows the maximum volume in each case
