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  • Level: GCSE
  • Subject: Maths
  • Word count: 1845

The Open Box Problem

Extracts from this document...

Introduction

The Open Box Problem

Equations for a square

Total length of paper = x

Total length of cutout = a

Width of base of box = c = x – 2a

Area of base = c2

Area of sides = 4ac

Volume = ac2

Square piece of paper with dimensions 12 x 12.

Cut out

Length

Width

Area of base

Area of sides

Volume

1

10

10

100

40

100

2

8

8

64

64

128

2.1

7.8

7.8

60.84

65.52

127.764

2.2

7.6

7.6

57.76

66.88

127.072

2.3

7.4

7.4

54.76

68.08

125.948

2.4

7.2

7.2

51.84

69.12

124.416

2.5

7

7

49

70

122.5

2.6

6.8

6.8

46.24

70.72

120.224

2.7

6.6

6.6

43.56

71.28

117.612

2.8

6.4

6.4

40.96

71.68

114.688

2.9

6.2

6.2

38.44

71.92

111.476

3

6

6

36

72

108

3.1

5.8

5.8

33.64

71.92

104.284

3.2

5.6

5.6

31.36

71.68

100.352

3.3

5.4

5.4

29.16

71.28

96.228

3.4

5.2

5.2

27.04

70.72

91.936

3.5

5

5

25

70

87.5

3.6

4.8

4.8

23.04

69.12

82.944

3.7

4.6

4.6

21.16

68.08

78.292

3.8

4.4

4.4

19.36

66.88

73.568

3.9

4.2

4.2

17.64

65.52

68.796

4

4

4

16

64

64

5

2

2

4

40

20

A graph to show the comparison of the cutout to the volume of a 12 x 12 square.

image00.png

Square piece of paper with dimensions 16 x 16.

Cut out

Length

Width

Area of base

Area of sides

Volume

1

14

14

196

56

196

2

12

12

144

96

288

2.1

11.8

11.8

139.24

99.12

292.404

2.2

11.6

11.6

134.56

102.08

296.032

2.3

11.4

11.4

129.96

104.88

298.908

2.4

11.2

11.2

125.44

107.52

301.056

2.5

11

11

121

110

302.5

2.6

10.8

10.8

116.64

112.32

303.264

2.7

10.6

10.6

112.36

114.48

303.372

2.8

10.4

10.4

108.16

116.48

302.848

2.9

10.2

10.2

104.04

118.32

301.716

3

10

10

100

120

300

3.1

9.8

9.8

96.04

121.52

297.724

3.2

9.6

9.6

92.16

122.88

294.912

3.3

9.4

9.4

88.36

124.08

291.588

3.4

9.2

9.2

84.64

125.12

287.776

3.5

9

9

81

126

283.5

3.6

8.8

8.8

77.44

126.72

278.784

3.7

8.6

8.6

73.96

127.28

273.652

3.8

8.4

8.4

70.56

127.68

268.128

3.9

8.2

8.2

67.24

127.92

262.236

4

8

8

64

128

256

5

6

6

36

120

180

...read more.

Middle

I have predicted that for a 20 x 20 square piece of paper that the maximum volume I can get will be with a cutout of 3cm.

Square piece of paper with dimensions 20 x 20

Cut out

Length

Width

Area of base

Area of sides

Volume

1

18

18

324

72

324

2

16

16

256

128

512

2.1

15.8

15.8

249.64

132.72

524.244

2.2

15.6

15.6

243.36

137.28

535.392

2.3

15.4

15.4

237.16

141.68

545.468

2.4

15.2

15.2

231.04

145.92

554.496

2.5

15

15

225

150

562.5

2.6

14.8

14.8

219.04

153.92

569.504

2.7

14.6

14.6

213.16

157.68

575.532

2.8

14.4

14.4

207.36

161.28

580.608

2.9

14.2

14.2

201.64

164.72

584.756

3

14

14

196

168

588

3.1

13.8

13.8

190.44

171.12

590.364

3.2

13.6

13.6

184.96

174.08

591.872

3.3

13.4

13.4

179.56

176.88

592.548

3.4

13.2

13.2

174.24

179.52

592.416

3.5

13

13

169

182

591.5

3.6

12.8

12.8

163.84

184.32

589.824

3.7

12.6

12.6

158.76

186.48

587.412

3.8

12.4

12.4

153.76

188.48

584.288

3.9

12.2

12.2

148.84

190.32

580.476

4

12

12

144

192

576

5

10

10

100

200

500

A graph to show the comparison of the cutout to the volume of a 19 x 19 square.

image02.png

The formula for the maximum volume of the box for square shaped pieces of paper is:

x = side of square

/ = divide by

x/6 = side of square / 6

...read more.

Conclusion

="c1">56

108

168

4

12

2

24

112

96

A graph to show the comparison of the cutout to the volume of a 10 x 20 rectangle.

image03.png

Rectangle piece of paper with dimensions 10 x 30.

Cut out

Width

Length

Area of base

Area of sides

Volume

1

28

8

224

72

224

1.1

27.8

7.8

216.84

78.32

238.524

1.2

27.6

7.6

209.76

84.48

251.712

1.3

27.4

7.4

202.76

90.48

263.588

1.4

27.2

7.2

195.84

96.32

274.176

1.5

27

7

189

102

283.5

1.6

26.8

6.8

182.24

107.52

291.584

1.7

26.6

6.6

175.56

112.88

298.452

1.8

26.4

6.4

168.96

118.08

304.128

1.9

26.2

6.2

162.44

123.12

308.636

2

26

6

156

128

312

2.1

25.8

5.8

149.64

132.72

314.244

2.2

25.6

5.6

143.36

137.28

315.392

2.3

25.4

5.4

137.16

141.68

315.468

2.4

25.2

5.2

131.04

145.92

314.496

2.5

25

5

125

150

312.5

2.6

24.8

4.8

119.04

153.92

309.504

2.7

24.6

4.6

113.16

157.68

305.532

2.8

24.4

4.4

107.36

161.28

300.608

2.9

24.2

4.2

101.64

164.72

294.756

3

24

4

96

168

288

4

22

2

44

192

176

A graph to show the comparison of the cutout to the volume of a 10 x 30 rectangle.

image04.png

Rectangle piece of paper with dimensions 10 x 40.

Cut out

Width

Length

Area of base

Area of sides

Volume

1

38

8

304

92

304

1.1

37.8

7.8

294.84

100.32

324.32

1.2

37.6

7.6

285.76

108.48

342.91

1.3

37.4

7.4

276.76

116.48

359.79

1.4

37.2

7.2

267.84

124.32

374.98

1.5

37

7

259

132

388.5

1.6

36.8

6.8

250.24

139.52

400.38

1.7

36.6

6.6

241.56

146.88

410.65

1.8

36.4

6.4

232.96

154.08

419.33

1.9

36.2

6.2

224.44

161.12

426.44

2

36

6

216

168

432

2.1

35.8

5.8

207.64

174.72

436.04

2.2

35.6

5.6

199.36

181.28

438.59

2.3

35.4

5.4

191.16

187.68

439.67

2.4

35.2

5.2

183.04

193.92

439.3

2.5

35

5

175

200

437.5

2.6

34.8

4.8

167.04

205.92

434.3

2.7

34.6

4.6

159.16

211.68

429.73

2.8

34.4

4.4

151.36

217.28

423.81

2.9

34.2

4.2

143.64

222.72

416.56

3

34

4

136

228

408

4

32

2

64

272

256

A graph to show the comparison of the cutout to the volume of a 10 x 40 rectangle.

image05.png

...read more.

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