• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month
Page
1. 1
1
2. 2
2
3. 3
3
4. 4
4
5. 5
5
• Level: GCSE
• Subject: Maths
• Word count: 1145

# Area &amp; Volume Exploration &amp;#150; Component proportional changes

Extracts from this document...

Introduction

Area & Volume Exploration – Component proportional changes

Question 2:          Suppose that you are required to design an open steel tray from a sheet of steel that measures 120cm by 80cm by cutting squares from each corner, and folding to form a tray. What size should the squares be cut from each corner of the sheet so that the maximum volume is obtained for the tray.

Height = x cm

Width  = 80cm - 2x cm

Length = 120cm – 2 x cm

The task is to find the optimal size of the squares that need to be cut out of the corners in order to find the maximum obtainable Volume inside the tray once it has been folded. To work this out there are a number of different methods for doing this. One method is Trial and Improvement.

Trial and Improvement

 Height (Value of xcm) 2 xcm Length (cm)(120cm – 2 x cm) Width (cm)(80cm - 2x cm) Volume (cm3)(120x  - 2 x 2)×(80 - 2x) +/- 5 10 110 70 38’500 7.5 15 105 65 51’187.5 + 10 20 100 60 60’000 + 12.5 25 95 55 65’312.5 += 15 30 90 50 67’500 + 17.5 35 85 45 66’937.5 -

Middle

+

15

30

90

50

67’500

+

16

32

88

48

67’584

+

=

17

34

86

46

67’252

-

The Volume starts to decrease after the value of 16 for x. So we know that the Maximum volume lies between 15 and 17. We can now continue the trial and improvement table working only between 15 and 17.

 Height (Value of xcm) 2 xcm Length (cm)(120cm – 2 x cm) Width (cm)(80cm - 2x cm) Volume (cm3)(120x  - 2 x 2)×(80 - 2x) +/- 15 30 90 50 67’500 15.25 30.5 89.5 49.5 67’561.3125 + 15.5 31 89 49 67’595 + 15.65 31.3 88.7 48.7 67’603.1485 += 15.75 31.5 88.5 48.5 67’602.9375 -

The table shows that the volume is at its greatest at 15.65 and decreases after this value. For increased accuracy  we can continue the trial and improvement table working between the x values of 15.50 and 15.8.

 Height (Value of xcm) 2 xcm Length (cm)(120cm – 2 x cm) Width (cm)(80cm - 2x cm) Volume (cm3)(120x  - 2 x 2)×(80 - 2x) +/- 15.500 31 89 49 67’595 15.525 31.05 88.95 48.95 67’597.44131 + 15.550 31.1 88.9 48.9 67’599.1155 + 15.575 31.15 88.85 48.85 67’600.52294 + 15.600 31.2 88.8 48.8 67’601.664 + 15.625 31.25 88.75 48.75 67’02.53.906 + 15.650 31.3 88.7 48.7 67’603.1485 + 15.675 31.35 88.65 48.65 67’603.49269 + 15.700 31.4 88.6 48.6 67’603.572 + = 15.725 31.45 88.55 48.55 67’603.38681 -

The optimal value of xto create a maximum volumehas now been narrowed down to between 15.675 and 15.725.

Conclusion

2. The maximum possible volume of the tray is 67’603.57731cm3

There are other methods that could be used to determine the maximum volume and the optimal value of X.One method could be differential calculus. This is applying the rules of differentiation to the object in question. This would be a quicker and more accurate method to determine the maximum volume and vale of X.

This investigation could be useful in engineering projects where a product must be produced with the maximum efficiency. This could mean being produced so that the volume inside is at its maximum possible and the wastage of material is at its least. Differentiation can also be used to determine dimensions of building sites or fields etc.

This exercises a very useful task that is widely used throughout many types of engineering and construction.

This student written piece of work is one of many that can be found in our GCSE Open Box Problem section.

## Found what you're looking for?

• Start learning 29% faster today
• 150,000+ documents available
• Just £6.99 a month

Not the one? Search for your essay title...
• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Related GCSE Open Box Problem essays

1. ## Maximum box investigation

700 512 324 400 44 Length of the side of the corner square Length of the box Width of the box Height of the box Volume of the box (cm�) 3.5 4.5 3.75 4.75 4.25 4.2 4.15 4.1 4.05 3.9 3.8 3.7 17 15 16.5 14.5 15.5 15.6 15.7 15.8

2. ## Open Box Problem

54.93564 1.2 2.6 17.6 1.2 54.912 1.3 2.4 17.4 1.3 54.288 1.4 2.2 17.2 1.4 52.976 1.5 2 17 1.5 51 2 1 16 2 32 2.5 0 15 2.5 0 Square cut in cm, volume in cm3 Rectangular Card Dimensions 10cm x 40cm 10 x 40 X cm Length

1. ## Tbe Open Box Problem

6 x must be 4.226 because 15.774 (3.d.p) would be more than the length. I shall prove this: Cut off (cm) Width (cm) Length (cm) Height (cm) Volume (cm�) 1 38 18 1 684 2 36 16 2 1152 3 34 14 3 1428 4 32 12 4 1536 5

2. ## THE OPEN BOX PROBLEM

2.18 192.297 5.62 15.62 2.19 192.248 5.60 15.60 2.20 192.192 As you can see by the table above, the largest volume is achieved when the cut out size of each corner of the box is 2.11cm. I also made a graph to prove that the maximum cut out size is around 2.11cm.

1. ## The open box problem

9.258974 0.829 3.342 3.342 0.829 9.259071 0.830 3.34 3.34 0.83 9.259148 0.831 3.338 3.338 0.831 9.259205 0.832 3.336 3.336 0.832 9.259241 0.833 3.334 3.334 0.833 9.259258 0.834 3.332 3.332 0.834 9.259255 0.835 3.33 3.33 0.835 9.259232 With a 5cm x 5cm piece of card we're able to see that a cut-out of 0.833 cm gives us the largest cut-out.

2. ## Investigate the volume of an open box constructed by one piece of rectangular card ...

length and height in terms of c, x and y yet this time I know that the length is exactly twice the width. Therefore I can replace these sub formulae into the first formula. I have constructed a table in excel where I can input the data for the cut

1. ## Trays.The shopkeepers statement was that, When the area of the base is the same ...

x 2 (n - 2 x X) x X (n - 2 x X) x X I take off two the corners from each side as the card is square. After finding out the formula I worked out the volume for the remaining trays.

2. ## The Open Box Problem.

to need to look at cut offs measuring between 3.3 and 3.4 cm. The table below shows the cut off measured to 2 decimal places. Looking at the table you should be able to see again the largest volume in bold, is with a cut out of 3.33cm.

• Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to