### The Open Box Problem

An open box is to be made from a sheet of card as shown below.

The corner squares are to be cut-off.The card is then folded along the dotted lines to make the box.—————————————————————————————————————————-Investigation 1 – Square shaped pieces of cardAim – To find the length of the cut-out corners that gives the maximum volume for the open box formed for any sized piece of square card. The length of the square cut will be to 3 significant figures of accuracy.Method – I will investigate what length of cut-out corners will give the largest volume ofr square pieces of card with dimensions 12 x 12, 18 x 18, 24 x 24 and 30 x 30.NOTE – when ‘small side’ is mentioned, it refers to the size of the cut-out corners.When ‘Length’, ‘Width’ and ‘Height’ are mentioned, they refer to the dimensions of the open box.

When ‘Volume’ is mentioned, it refers to the volume of the open box.Rows in Italics are those which contain the correct cut-out corner size for the maximum volume of the open box.Square piece of card with dimensions 12 x 12Small SideVolumeLengthWidthHeight110010101212888231086632.1127.7647.87.

82.12.2127.0727.67.62.22.3125.

9487.47.42.32.4124.

4167.27.22.42.5122.5772.51.

9127.7568.28.

21.91.8127.

0088.48.41.81.95127.

93958.18.11.951.99127.99768.

028.021.99Graph comparing the length of Small Side to the Volume for a square shaped piece of card with dimensions 12 x 12Square piece of card with dimensions 18 x 18Small SideVolumeLengthWidthHeight2392141423432121234400101043.

5423.511113.53.

6419.90410.810.83.63.

4426.49611.211.

23.43.3428.86811.411.43.33.

2430.59211.611.63.23.

1431.64411.811.83.13.09431.

7113211.8211.83.093.08431.7716511.8411.

83.083.05431.910511.911.93.053.

03431.9677111.9411.93.033.02431.9856311.

96123.023.01431.996411.98123.01Graph comparing the length of Small Side to the Volume for a square shaped piece of card with dimensions 18 x 18Square piece of card with dimensions 24 x 2428002020239721818341024161643.51011.

517173.53.61016.06416.816.83.

63.71019.57216.616.63.73.

81022.04816.416.43.83.91023.51616.216.

23.93.911023.608316.1816.23.

913.921023.690816.1616.23.923.

931023.763416.1416.

13.933.941023.826316.1216.

13.943.951023.879516.

116.13.953.961023.922916.

0816.13.963.971023.956716.0616.13.973.

981023.980816.04163.983.

991023.995216.02163.99Graph comparing the length of Small Side to the Volume for a square shaped piece of card with dimensions 24 x 24In each previous case, the length of the cut-out corner squares has benn 1/6 of the side of the square pieces of card.

I predict that for a square piece of card 30 x 30, the side of the small square cut-outs wil be 5 cm.Square piece of card with dimensions of 30 x 30Small SideVolumeLengthWidthHeight3172824243419362222452000202054.51984.521214.54.61990.14420.

820.84.64.

71994.49220.620.

64.74.81997.

56820.420.44.84.91999.39620.220.

24.94.911999.511120.1820.24.914.

951999.849520.120.14.954.

991999.99420.02204.99Graph comparing the length of Small Side to the Volume for a square shaped piece of card with dimensions 30 x 30Formula to give the maximum open box volume for square shaped pieces of cardx = Side of square shaped piece of cardx/6 = side of cut-out cornerVolume = Length x Width x Height__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________—————————————————————————————————————————-Investigation 2 – Rectangular shaped pieces of cardAim- To find the length of the cut-out corner squares that give the maximum open box volume for rectangular pieces of card of different sizes. The length of the cut-out corner squares will be to 3 significant figures of accuracy.

Method- I will investigate the size of the corner cut-out squares that give the largest open box volume for rectangular shaped pieces of card that have the width to length ratio of 1:2, those being 12 x 24, 24 x 48 and 48 x 96. I will then produce a formula for the maximum open box volume, for all rectangles that have width to length ratio’s of 1:2. I will also investigate the size of the corner cut-out squares that give the largest open box volume for rectangular shaped pieces of card that have the width to length ratio of 1:3, those being 5 x15, 15 x 45 and 45 x 135..I will then produce a formula for the maximum open box volume, for all rectangles that have width to length ratio’s of 1:3. Eventually I will produce a formula that gives the maximum open box volume, for all rectangles. The size of the cut-out corners will be to 2 decimal places of accuracy or 3 significant figures of accuracy.NOTE – when ‘small side’ is mentioned, it refers to the size of the cut-out corners.When ‘Length’, ‘Width’ and ‘Height’ are mentioned, they refer to the dimensions of the open box.When ‘Volume’ is mentioned, it refers to the volume of the open box.Rows in Italics are those which contain the correct cut-out corner size for the maximum volume of the open box.Rectangular piece of card with dimensions 12 x 24Small SideVolumeLengthWidthHeigth23202082332418632.5332.51972.52.4331.77619.27.22.42.6332.38418.86.82.62.55332.545518.96.92.552.54332.5530618.926.922.542.56332.5296618.886.882.562.57332.5055718.866.862.572.58332.4732518.846.842.582.53332.552318.96.92.532.52332.5432318.966.962.522.51332.525818.986.982.512.49332.465819.027.022.49Graph comparing the length of Small Side to the Volume for a rectangular shaped piece of card with dimensions 12 x 24Rectangular piece of card with dimensions 24 x 48Small SideVolumeLengthWidthHeigth3226842183425604016452660381455.12660.36437.813.85.15.22659.07237.613.65.25.152659.923537.713.75.155.132660.149237.7413.75.135.122660.237337.7613.85.125.112660.308937.7813.85.11Graph comparing the length of Small Side to the Volume for a rectangular shaped piece of card with dimensions 24 x 48Rectangular piece of card with dimensions 48 x 96Small SideVolumeLengthWidthHeight82048080328921060783091021280762810112116474261110.521262.5752710.510.621249.18474.826.810.610.421272.57675.227.210.410.321279.38875.427.410.310.221282.91275.627.610.210.121283.12475.827.810.110.2421281.89875.5227.510.2410.2221282.47175.5627.610.2210.2121282.70875.5827.610.21Graph comparing the length of Small Side to the Volume for a rectangular shaped piece of card with dimensions 48 x 96In each case above where the ratio of the sides of rectangles is 1:2, the length of small side that has given the largest volume has been the width/4.7Formula to give the maximum open box volume for rectangular shaped pieces of card with the ratio of 1:2x = width of rectangular shaped piece of cardx/4.7 = side of cut-out cornerVolume = Length x Width x Height__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________Rectangular piece of card with dimensions 5 x 15Small SideVolumeLengthWidthHeight139133122211121.5361221.51.437.57612.22.21.41.338.68812.42.41.31.239.31212.62.61.21.139.42412.82.81.11.1539.433512.72.71.151.1439.44217612.722.721.141.1339.44558812.742.741.13Graph comparing the length of Small Side to the Volume for a rectangular shaped piece of card with dimensions 5 x 15Rectangular piece of card with dimensions 15 x 45Small SideVolumeLengthWidthHeight290241112310533993410363774587535553.510643883.53.41065.01638.28.23.43.31064.44838.48.43.33.451064.704538.18.13.453.441064.798338.128.123.443.431064.876438.148.143.433.421064.938838.168.163.423.411064.985338.188.183.41Graph comparing the length of Small Side to the Volume for a rectangular shaped piece of card with dimensions 15 x 45Rectangular piece of card with dimensions 45 x 135Small SideVolumeLengthWidthHeight7262571213178276081192989284311172791028750115251010.5287281142410.510.428741.856114.224.210.410.328751.008114.424.410.310.228755.432114.624.610.210.128755.104114.824.810.1Graph comparing the length of Small Side to the Volume for a rectangular shaped piece of card with dimensions 15 x 45In each of the above cases the length of the small side that gives the maximum open box volume has been width/4.4Formula to give the maximum open box volume for rectangular shaped pieces of card with the ratio of 1:3x = width of rectangular shaped piece of cardx/4.4 = side of cut-out cornerVolume = Length x Width x Height__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________—————————————————————————————————————————-Formula to give the maximum open box volume for rectangular shaped pieces of card with the any ratiox = width of rectangular shaped piece of cardx/ = side of cut-out cornerVolume = Length x Width x Height