# maximum length of rectangles and volume

Extracts from this document...

Introduction

Ben Patience Maths Coursework

Conclusion

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.

Middle

‘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.

Conclusion

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.

This student written piece of work is one of many that can be found in our GCSE Comparing length of words in newspapers section.

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