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• Level: GCSE
• Subject: Maths
• Word count: 1974

# Mathematics Gcse Coursework Tubes Investigation

Extracts from this document...

Introduction

Mathematics GCSE Coursework

## Syllabus 1385

TUBES

Objective:The coursework is separated into three investigations. The first investigation is to investigate the volumes of open-ended tubes, which can be made from a 24cm by 32 cm rectangular piece of card. The second investigation is to investigate the volume of open-ended tubes, which can be made from any rectangular piece of card. The third investigation is that for a given area of a rectangular piece of card; investigate the volumes of opened tubes, which can be made.

## Investigation No.1

The aim for investigation no.1 is to investigate the volumes of open-ended tubes, which can be made from a 24cm by 32cm rectangular piece of card. There are two methods that the tubes can be made.

Take an open-ended cube for example:

The first method that cube can be construct as the height 24cm with the length and width 8cm. This means the cube is constructed using the paper landscape and way the cube will look short and wider.

The second method that the cube can be constructs having a height of 32cm with the length and the width 6cm. This means the cube is constructed using the paper portrait and way the cube will look higher and thinner.

After using the formula: height x width x length to calculate the volume of cube.

Middle

24 ÷ 6 = 4cm           length of one side

4 ÷ 2 = 2cm            half of one side

2 × tan 60 degree = 3.5cm          the height of one triangle

4 × 3.5 = 14cm2          area of a square

14 ÷ 2 = 7cm2         area of a triangle

7 × 6 = 42cm2         area of the whole hexagon

42 × 32 = 1344cm3         the volume of the hexagon tube

## Cylinder

 Area = Circumference =

All the results presented in the table below:

 Different types of tubes The volume of tubes constructed with the paper landscape  (cm3) The volume of tubes constructed with the paper portrait (cm3) Triangular Prism 1194 883.2 Cube 1536 1152 Hexagon 1755 1344 Cylinder 1961 1451.7

Conclusion for Investigation No.1

To conclude investigation 1, for any tubes constructed with either using the paper landscape or portrait, the biggest volume found is when the tube has more sides but eventually, after i.e. when the tube have infinity sides then the tube will then become a cylinder so the biggest volume found is when it is a cylinder. And also the tubes constructed with the paper landscape are always bigger in volume than the tubes constructed with the paper portrait.

### Investigation No.2

The aim for investigation no.2 is to investigate the volume of open-ended tubes, which can be made from any rectangular piece of card. Method no.1 is used for this investigation because I found out those short and wide tubes holds a larger volume than the high and thin tubes in investigation no.1.

Take an open-ended cube for example:

If the piece of rectangular paper is 24cm by 32cm, to construct a cube by using the paper landscape then the height will be 24cm with the length and width 8cm. The volume of this cube is 1536 cm3. But what will happen if the 24cm by 32cm paper was cut into half horizontally, given the cube’s height 12cm with the length and the width 16cm? And what will also happen if I cut the 24cm by 32cm paper in half horizontally two times given the cube’s height 8cm with the length and width 24cm and then what will happen if I cut the 24cm by 32cm paper three times and so on. What effect does it have on the volume if the height of the cube is lower with the width and length longer?

The diagrams below will help you to have a better understanding of this investigation:

 The piece of paper is 24cm by 32cm. To calculate the volume of this cube then must follow the steps:32 ÷ 4 = 8cm               the length of one side8 × 8 = 64cm2               the area of the surface area of the cube64 × 24 = 1536cm3                the volume of the cube The paper has cut into half horizontally and stuck together. The width is two times longer and the length is two times shorter. Given the piece of paper is 12cm by 64cm. To calculate the volume of this cube then must follow the steps:64 ÷ 4 = 16cm16 × 16 = 256cm2256 × 12 = 3072cm3
 The 24cm by 32cm paper has cut three times and stuck together. The width is three times longer and the length is three times shorter. Given the piece of paper is 8cm by 96cm.To calculate the volume of this cube then must follow the steps:96 ÷ 4 = 24cm24 × 24 = 576cm2576 × 8 = 4608cm3 The 24cm by 32cm paper has cut four times and stuck together. The width is four times longer and the length is four times shorter. Given the piece of paper is 6cm by 128cm. To calculate the volume of this cube then must follow the steps:128 ÷ 4 = 32cm32 × 32 = 1024cm21042 × 6 = 6144cm3 The 24cm by 32 cm paper has cut five times and stuck together. Imagine the width now is five times longer and the length is five times shorter. Given the piece of paper is 4.8cm by 160cmTo calculate the volume of this cube then must follow the steps:160 ÷ 4 = 40cm40 × 40 = 1600cm21600 × 4.8 = 7680cm3

Conclusion

We know the triangle is an isosceles triangle so the two bottom angles are the same and 180 degrees is the total sum for a triangle thus 180 – the top angle then divide by two then you will get the bottom two angles and the bottom two angles should be identical. The following step is to multiply tangent the top angle by the length of half a side. After this, you can find the height of the triangle.

Then multiply the height of triangle to the length of side then you will find the area of the square. Then divide it by 2 to get the area of the triangle. Then multiply the area of triangle to the ‘n’ side to find the total area of the polygon. Afterward, multiply the area of polygon to the height of tube to find the whole volume of the polygon tube.

Here is the formula that can calculate the volume of polygon tube for any given area of a rectangular piece of paper

But however this formula can be simplify:

## Formula of Cylinder

The area of circle is =

The circumference is =

But buried in mind, the circumference of the circle is the width of the paper so

can be written as       = w

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