Tube Maths Investigation

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For this project we had to investigate the volumes of open-ended tubes made from a rectangular piece of card with the dimensions, 32cm by 24cm. Either of the two sides could be folded to make the perimeter, with the other side being the height. I made a couple of conjectures before I started. They were:

· the more sides, the bigger the volume

· regular polygons are the best


1. For the first question, the volumes have to be investigated from various polygon shapes as the base side.

The first shape I used was a square, with a height of 24cm, and perimeter of 32cm:



To find the area of the base, I used the formula: A = x2, where ‘x’ is the side of the square. Once the area was found it was multiplied by the height.

A = 82

V = 1536cm3

For a square with height 32cm and perimeter 24cm, it is:

A = 62

V = 1152cm3

From the method I used, a formula can be found for an open ended tubes with square bases. If we let ‘h’ be the height, and ‘p’ be the perimeter, it would be:

V = p 2 x h

4

V = p 2 x h

16

V = p2h

16

The next shape was an equilateral triangle based tube:

2 3

10h = 24

cm

1 2To find the base area the formula: A = b x h x , with ‘b’ the base of the triangle, and ‘h’ the height.

1 2A = 10.66 x v(10.662 – 5.332) x

V = 1182cm3

This is the result for a triangle with height of 32cm, and perimeter of 24cm:

1 2A = 8 x v(82 – 42) x

V = 887cm3

From this a formula could be established:

1  2V = p x v p 2 - p 2 x x h

3 3 6

V = p x v p2 xh

6 12

The next shape I used as a base was a hexagon.

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each of the sides has a length of 4cmh = 32

For this, the hexagon was split up into 6 equilateral triangles, with the area of one of those found then multiplied by 6 (the number of triangles). This meant the same formula was used as with the triangle, but multiplied by 6.

1  2A = 4 x v(42 – 22) x x 6

V = 1330cm3

Using the other dimensions with 24cm the height, the volume was:

1  2A = 5.33 x v(5.332 – 2.662) x x 6

V = 1773cm3

The formula was deduced from the method, which ...

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