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maximum length of rectangles and volume

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Ben Patience                Maths Coursework


When I began this take of trying to find out the maximum length of rectangles in different my initial task was simply to find a formula to relate to the rectangles. To solve the problem I used computer programmes such as Microsoft Excel and Microsoft Word to present the data and my findings. I initially chose to work with squares 8x8, 9x9, 13x13, and 25x25 for the simple fact that I was looking for some connection between all types of squares.

When I used the computer I headed my columns on my spreadsheets as follows:

Side1 (length), Side2 (width), Cut out, Length, Width, Height, Volume. I used these in this order to try and determine the formula and maximum cut out length for each type of square and rectangle.

To work out different volumes I used different formulas. I worked out these formulae by simple trial and error. This did take some time but eventually came out more accurate.

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‘Y’ axis: I used the size of the rectangles to show the size of each rectangle volumes against,

‘X’ axis: I used the size of the cut out length to relate it to the size of the rectangle.

The formula to find the volume of a box is:

Size 1         = a

Size 2          = b

Cut-out        = c

Length        = a – 2c

Width         = b – 2c

Height         = c

Volume = Length x Width x Height

Volume = (a – 2c) x (b – 2c) x c

Volume = ab – 2ac -2ab + 4c

When searching for a rule for the different ratios of rectangles I noticed that each ratio had different formulae for which I had to discover each one by either trial and error or by plotting a graph as shown below.

When using excel now I know how to input formulas to make easier equations and to show them with a plotted graph with a trend line and the formula with it. This makes life a lot easier because otherwise I would have had to use trial and error.

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30 x 0.10565 = 3.1695

This means that 3.1695 is the cut out length that would give the maximum volume for a 15x30 rectangle.

However in order to use this method you first must be able to work out the maximum volume of at least 2 rectangles in the same ratio to be able to plot the graph. This is essential because you cannot plot a graph for one measurement. Finally, the more the measurements that you make for the ratio of 1:2 the more accurate that the graph is going to be.

If you look on my formulas page you will notice that I have all the equations for the ratios 1:2, 1:3 and 1:4 along with the squares formula which I have added above.

If I had more time I would create more squares and make it to 4 decimal places. This would increase the chance that my maximum volume would be bigger and be more precise to the maximum cut out length. I would also test more ratios and try to find a link between greater ratios a find a formula to make the ratios link to each other.

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