GCSE: Open Box Problem
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 Level: GCSE
 Questions: 75

Open Box Investigation
5 star(s)The height is the same as the side length of square cut. Below are my results: 10x10 Square Side Length of Square (cm) 1 2 3 4 Length of Box 8 6 4 2 Width of Box 8 6 4 2 Height 1 2 3 4 Volume 64 72 48 16 To find the more precise length of square which I should cut, I have used the upper and lower bounds of the best length of square in the table above (in this case, 2cm). Side Length of Square 1.5 1.6 1.7 1.8 1.9 2 2.1 Length of Box 7 6.8 6.6 6.4 6.2 6 5.8 Width of Box 7 6.8 6.6
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Maths Coursework
3 star(s)The squares of sizes I will look at are: 12cm x 12cm, 15cm x 15cm and 18cm x 18cm. The first square I will look at is 12cm by 12cm. Size of square cut out (Height x) Length (122x) Width (122x) Volume (cm�) L x W x H 1 10 10 100 2* 8 8 128 3* 6 6 108 4 4 4 64 5 2 2 20 2.5 7 7 122.5 2.6 6.8 6.8 120.224 2.4 7.2 7.2 124.416 2.3 7.4 7.4 125.948 2.2 7.6 7.6 127.072 2.1* 7.8 7.8 127.764 2* 8 8 128 1.9 8.2 8.2 127.756 2.05* 7.9 7.9 127.9405 The cut out square which proved to have the biggest volume was 2cm by 2cm.
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Applications of Differentiation
This can be solved using trial and error by forming a table and trying different possibilities of the squares that will be cut. Size of square Length of cuboid Width of cuboid Height of cuboid No. of cubes 1 x 1 18 10 1 180 2 x 2 16 8 2 256 3 x 3 14 6 3 252 4 x 4 12 4 4 192 Looking at the 1 x 1 square; if a 1 x 1 square is removed for the corners, the length becomes 20 1 1 =18cm.
 Word count: 657

Maximum box investigation
Results Part 1 At first I only tried out the length of the corner square being a whole number. As you can see from this table (below), I found that if the length of the corner square was 3 then the volume of the box would be the greatest. However I wanted to research this further and see if the volume could be larger if the corner squares length also had a decimal number. In the table at the bottom of the page I have showed my results.
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Open Box Problem
This was for the square being removed size 1cm, and it was the same formula for length and width as it is a square. The depth is always the same as the size of square cut (x). I then multiplied the length, depth and width to give me the volume. With the 10 x 10 square, the largest volume so far was between 1.5cm and 2.5cm, so I then put in square cut sizes at an interval of 0.1 cm between 1.5 and 2.5.
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Maths Courseowrk  Open Box
You will see this process repeated four times for each of my tested squares. After doing these stages I will try and work out a relationship between the cut out of the square and the volume. To do this I will generalise the formula then use a method called differentiation. After I have done this I will input the letters and numbers into the quadratic formula and work out the relationship. Throughout this piece of coursework I will be using this formula v=x(l2x)
 Word count: 1918

Tbe Open Box Problem
Cut off (cm) width (cm) length (cm) height (cm) volume (cm�) 1.61 6.78 6.78 1.61 74.0091 1.62 6.76 6.76 1.62 74.0301 1.63 6.74 6.74 1.63 74.047 1.64 6.72 6.72 1.64 74.0598 1.65 6.7 6.7 1.65 74.0685 1.66 6.68 6.68 1.66 74.0732 1.67 6.66 6.66 1.67 74.0739 1.68 6.64 6.64 1.68 74.0705 1.69 6.62 6.62 1.69 74.0632 The highest value of x is therefore 1.67. I will now go into even smaller numbers between 1.66 and 1.67. Cut off (cm) width (cm)
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The Open Box Problem
125.948 7.4 7.4 2.3 2.4 124.416 7.2 7.2 2.4 2.5 122.5 7 7 2.5 1.9 127.756 8.2 8.2 1.9 1.8 127.008 8.4 8.4 1.8 1.95 127.9395 8.1 8.1 1.95 1.99 127.9976 8.02 8.02 1.99 Graph comparing the length of Small Side to the Volume for a square shaped piece of card with dimensions 12 x 12 Square piece of card with dimensions 18 x 18 Small Side Volume Length Width Height 2 392 14 14 2 3 432 12 12 3 4 400 10 10 4 3.5 423.5 11 11 3.5 3.6 419.904 10.8 10.8 3.6 3.4 426.496 11.2 11.2
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Global warming is an important factor because every one in the world but most importantly for me it will effect all of my relatives, as it is predicted that countries such as Bangladesh will be totally fooled due to global warming.
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The Open Box Problem
I found that the highest volume must have a cut out size of between 1 and 2cm so I tried the formula for cut out sizes between 1 and 2cm: size of cut out 'x' (cm) volume (cm�) 1.1 66.924 1.2 69.312 1.3 71.188 1.4 72.576 1.5 73.5 1.6 73.984 1.7 74.052 1.8 73.728 1.9 73.036 I again found that the highest volume is 74.052cm� with a cut out size of 1.7cm. I then homed in again and used the formula for cut out sizes between 1.6 and 1.7cm.
 Word count: 2232

THE OPEN BOX PROBLEM
I am using a length/width of 10cm. I am going to call the cut out "x." Therefore the equation can be changed to: Volume = 10  (2x) * 10  (2x) * x If I were using a cut out of length 1cm, the equation for this would be as follows: Volume = 10  (2 * 1) * 10  *(2 * 1) * 1 So we can work out through this method that the volume of a box with corners of 1cm� cut out would be: (10  2) * (10  2)
 Word count: 3128

The open box problem
The same formula is then used for calculation of depth. Note: When putting formulas in for real you always put an = sign before typing a formula, Excel then knows your typing in a formula and will work out the answer to it when finished. Not done on above as wanted to show what the formulas were. Results and Evidence These are the results for a 6cm x 6cm piece of card. Max Box Size of card 6 Size of cutout Width Depth Height Volume 0 6 6 0 0 1 4 4 1 16 2 2 2 2 8 3 0 0 3 0 Size of cutout Width Depth
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Investigate the volume of an open box constructed by one piece of rectangular card that has all four corners having had squares cut out of them.
card, I will now need to show the values of the width, length and height in terms of c, x and y. Therefore I can replace these sub formulae into the first formula. However for the values of a square, y=x therefore: c V 1 324 2 512 3 588 4 576 5 500 6 384 7 252 8 128 9 36 As I have already said, the value of x is 20cm and I will use a range of values for c, then I will plot a graph with c on the xaxis and the volume (V)
 Word count: 2481

Boxed In.
I will work between the cut outs 3 and 4 centimetres. To get the Length/Breath of the box; to work out the volume in my trial and improvement table; I will double the cut out number and subtract it from 20. E.g. Cut out = 3.1 20  (3.1 + 3.1) = Length/Breath I will then take the length and square it (breath is the same as the length) and multiply it by the height; which is the size of the cut out. E.g. 13.82 x 3.1 = 590.36 (the total volume of the box)
 Word count: 2540

Trays.The shopkeepers statement was that, When the area of the base is the same as the area of the four sides, the volume of the tray will be maximum.
(The longest possible corner could only be 8 as after this there would be no base.) After this I worked out the formula needed to work out the volume for the various trays. For the corner size 1x1 the way I worked out the volume was 16x16x1 which equalled 256cm. Thus the formula to work out the volume for a tray made by an 18x18cm card is (n  2X) x X. In this formula the letter "X" represents the size of the corner. I tried my formula for the corner length of 2cm, (18 2 x 2)
 Word count: 1391

Kunstnik ja dekoraator.
Detailid on need, mis ekraanimaailma elusaks muudavad ja vaataja �mbritsevat unustama panevad. Siin oli aga tegu �he suure teleteatriga. Arvan, et mina ei olnud ainuke, kellel n�iteks stseenis, kus Ahast taga aetakse, tekkis kriipiv ja kahtlustav tunne , mis �tleb:"Vaata, vot selle aia taga seisab uus mersu ja tolle kuuri taga RKiosk...". No ei suudetud seda ajastu h�ngu filmi sisse tuua ja k�ik. V�ibolla on viga miseenscene`is. Tihti tuli ette olukordi, kus esiplaanil olevat tegevust saatis tagaplaanil olev kohatu t�hjus.
 Word count: 1069

The Open Box Problem.
X = ^1/[6]A 4m by 3cm, piece of card. R e s u l t s Length of the section (cm) Height of the section (cm) Depth of the section (cm) Width of the section (cm) Volume of the cube (cm^3) 0.1 0.1 2.8 3.8 1.064 0.2 0.2 2.6 3.6 1.872 0.3 0.3 2.4 3.4 2.448 0.4 0.4 2.2 3.2 2.816 0.5 0.5 2.0 3.0 3.000 0.51 0.51 1.98 2.98 3.0092 0.52 0.52 1.96 2.96 3.0168 0.53 0.53 1.94 2.94 3.0229 0.54 0.54 1.92 2.92 3.0275 0.55 0.55 1.90 2.90 3.0305 0.56 0.56 1.88 2.88 3.0321 0.57 0.57 1.86 2.86
 Word count: 2760

Open Box Problem.
x Width  (2 x Cut Out) x Height OR This therefore is the general formula. X = size of cutout and height L = length of square sheet Volume = X(L2X)(L2X) X = X(L2X)^2 [image004.gif] [image005.gif] [image006.gif] [image007.gif] [image008.gif] ^ 20cm 20 cm Height X Width L2X Length L2X We get this formula by multiplying the width and length and then multiplying by the height If I were using a cut out of length 1cm, the equation for this would be as follows: Volume = 1(202x1)^2 So we can work out through this method that the volume of a
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Investigation: The open box problem.
�x3 V = 64 V = 72 V = 48 X = 4 X = 5 V = (10(2x4)) �x4 Not possible due to reasons mentioned above. V = 16 If five was cut from each corner then there would be nothing left to make the box with. Graph To show the cut off compared to the volume. The cut off that gives the largest volume is 2. I will now look for the cut off that will leave the largest volume for the box. I will be looking for the cut off to 1d.p. between 1.5  2.5 X = 1.5 X = 1.6 X = 1.7 V = (10(2x1.5)) �x1.5 V = (10(2x1.6)) �x1.6 V = (10(2x1.7))
 Word count: 11760

In this coursework my task description is to see how fast I can catch a reaction time ruler. I have to do this with both my hands and see the difference between the hands.
19 17 18 15.5 15 16.5 15 20 18 12 15 13 18 17 18 14 20 21 19 17 16 17 12 16 12 10 21.5 18.5 19 13 17.5 11 17.5 17.5 18 16 20 15 21 13 15 16 17 4 16 13 14 13 21 14 17.5 13 16 19 14 15 GIRL 1 GIRL 2 GIRL 3 RIGHT HAND LEFT HAND RIGHT HAND LEFT HAND RIGHT HAND LEFT HAND 23 20 17.5 12 15.5 11 12 21 15 13 15 23 19 23 23 14 11 19 20 17 14.5 15.5 22 9 5 8
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Open box Problem.
Now I am going to find out the volume when X is between 3.5 and 4.5. Here are the results: X(cm) X(24cm2X)(cm) 3.5 1011.5 3.6 1016.064 3.7 1019.572 3.8 1022.048 3.9 1023.516 4.0 1024 4.1 1023.524 4.2 1022.112 4.3 1019.788.4.4 4.4 1016.516 4.5 1012.5 We found out that 4cm is still the biggest value whereas we can get the biggest volume. Now I am going to set the length to be 12. With a cut out of 1cm,X=1cm, 5cm is the maximum for there to be a box left.
 Word count: 2711

Open Box Problem
of 2cm by 2cm. Once the cutoffs are taken away, the net will look like this. From this we can see that when the dotted lines are folded along, there is a height of 2cm, a length of 16cm and a width of 16cm. Since Volume = Length * Width * Height, the volume of this open box is 16*16*2 = 512cm3 We can also see relationships between the cutoff and the dimensions of the net from this example. 1.
 Word count: 1817

The Open Box Problem
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 12x12 square X=1 10x10x1 100 cm X=2 8x8x2 128 cm X=3 6x6x3 108 cm X=4 5x5x4 100 cm X=5 4x4x5 80
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The Open Box Problem
Finally I will draw a conclusion for both the square card and the rectangular card. The Square I will first use a 10 x 10 square S X B V 10 1 8 64 10 2 6 72 10 3 4 48 10 4 2 16 10 5 0 0 10 6 2 24 10 7 4 112 10 8 6 288 10 9 8 576 10 10 10 1000 S X B V 10 1 8 64 10 1.1 7.8 66.924 10 1.2 7.6 69.312 10 1.3 7.4 71.188 10 1.4 7.2 72.576 10 1.5 7 73.5 10 1.6
 Word count: 3299