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Maths IA Type 2 Modelling a Functional Building. The independent variable in this investigation is the height of the building. The maximum volume of a cuboid under the roof depends on the height of the roof, which is the dependant variable.

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Introduction

Maths IA Type 2                Francis Nguyen

QUEENSLAND ACADEMY FOR SCIENCE, MATHS AND TECHNOLOGY

Maths IA Type 2

Modelling a Functional Building

Francis Nguyen

Mr Mathews


Introduction:

The structure of a roof for a building is parabolic. The design of this building has a fixed rectangular base which is 150m long and 72m wide. The maximum height of this building can vary between 50% - 75% of its width for stability and aesthetic purposes. The independent variable in this investigation is the height of the building. The maximum volume of a cuboid under the roof depends on the height of the roof, which is the dependant variable.

All calculations will be made through Ti-nSpire calculator (GDC; Graphical Display Calculator) and all figures will be rounded to 3 decimal places as architects work with millimetres where as this report works in meters.

The Function:

The model of the roof structure will be designed on a Cartesian plane using the graphing package, Graph. The axis of symmetry (also the maximum turning point in this case) will be modelled at image16.pngimage16.png within this report (refer to Fig 1 below). The roots of the quadratic will be at a fixed co-ordinate of          (-36,0) and (36,0) as the distance between these 2 points is 72m long for when the façade is designed at the width (refer to Fig 2).

...read more.

Middle

image34.png value)

Height of cuboid (m)   (image17.pngimage17.png value)

Length of cuboid (m)

Max volume of cuboid (m3)

36

image35.pngimage35.png

41.569

24

150

149649.190

45

image36.pngimage36.png

41.569

30

150

187061.487

52

image37.pngimage37.png

41.569

34.667

150

216162.019

53

image38.pngimage38.png

41.569

35.333

150

220314.784

54

image39.pngimage39.png

41.569

36

150

224473.785

Through inspection, the width remains the same for all tested heights. It is evident that as the height of the roof increases by 1m, the height of the also cuboid increases, by image41.pngimage41.pngm. For this type of model function, the height of the cuboid can be found by image42.pngimage42.png; where image17.pngimage17.png is the height of the cuboid and image43.pngimage43.png is the maximum height of the roof.

Method 4. Finding the increase in height

Let image44.pngimage44.png

Using values of roof height at 52m and 45m

image45.png

image46.png

image47.png

The width of the maximum volume of the cuboid remains the same, the length is fixed at 150m, and the height of the cuboid increases as the maximum height of the roof increases. Therefore, the volume should also increase in proportion to the height of the roof as image48.pngimage48.png.

Ratio of Wasted Space to Office Block:

After calculating the maximum volume of the cuboid, it is important to determine how efficient the volume of the cuboid is by comparing it to the volume of wasted space. The volume of wasted space is found by, image49.pngimage49.png.

Method 5:

To find volume under the curve algebraically, we integrate:

image50.png

Using GDC to find the answer:

image51.png

Fig 4 below shows the area under the curve which was calculated. The area  under the curve was also determined by using Graph which supports the answer that was found.

Fig 5. Area under the graph (and between the image53.pngimage53.png axis)image02.png

image55.pngimage54.png

Multiply by length to find the volume:

image56.pngimage56.png

image57.png

image58.pngimage58.png

image59.png

image60.png

The following table was constructed by following the same methodology in Method 4, and the maximum volume was obtained from table 1.

Table 2. How the maximum height of the roof affects ratio of volume wasted

Maximum height of roof (m)

Volume of wasted space (m3)

Max volume of cuboid (m3)

Ratio

Wasted space : Volume cuboid

36

109550.810

149649.190

0.732 : 1

45

136938.513

187061.487

0.732 : 1

52

158237.981

216162.019

0.732 : 1

53

161285.216

220314.784

0.732 : 1

54

164326.215

224473.785

0.732 : 1

Table 2 above shows that ratio of wasted space to the volume of the cuboid does not change as the height varies. This tells us that the efficiency of the space cannot be improved by varying the height. The table also tells us that approximately the value of 73.2% of the volume of the cuboid is equivalent to the wasted space.

Maximum Cuboid Floor Area:

We now investigate the maximum cuboid floor area to determine how much space we are able to use. Each level of the cuboid is at a fixed height of 2.5m.

Method 6. Finding the maximum floor area of the cuboid

The office floor area of one level is:

image61.png

As the length and width are at fixed dimensions of 150m and 41.569m (or image62.pngimage62.pngm)

image63.png

   The area of each floor is:

image64.png

The number of levels is found by:

image65.pngimage65.png ; where image17.pngimage17.png is the height of the office block

∴   The total floor area can be found by:

image67.png

Method 7. Finding the maximum floor level when the roof height = 36m

At image68.pngimage68.pngm

image69.png

image70.png

image71.pngimage71.png

Cannot contain 10 floors, but can contain 9 floors. Therefore we round the number of levels to the lowest positive integer:

image72.png

image73.png

Using Method 7, the following table was constructed.

Table 4. Maximum area of cuboid floor:

Maximum height of roof (m)

Number of levels

Maximum area of cuboid floor (m2)

36

9

56118.447

42

16

99766.128

46

18

112236.894

52

20

124707.660

53

21

130943.043

54

21

130943.043

Switching the Side of the Façade

In order to test for more efficient possibilities, the face of the façade will be on the longer side of the base, where the length is 150m.

Using the same methodology in Method 1, we obtain the standard formula when the façade is on the length:

image74.png

Fig 6. Model of roof structure at minimum height when the façade switches

image75.pngimage03.pngimage04.pngimage07.pngimage05.pngimage06.png

Using the methodology shown in Method 3, the follow table was constructed, where the maximum and minimum height was found. 2 consecutive heights near the maximum, and a few points near the medium was found.

Table 5. dimension and maximum volume of cuboid when the façade switches

Maximum height of roof (m)

Function

image76.pngimage76.png

Width of cuboid (m)  

Height of cuboid (m)

Length of cuboid (m)

Max volume of cuboid (m3)

75

image77.pngimage77.png

86.603

50

72

311769.145

85

image78.pngimage78.png

86.603

56.667

72

353340.443

95

image79.pngimage79.png

86.603

63.333

72

394905.506

110

image80.pngimage80.png

86.603

73.333

72

457259.335

111

image83.pngimage83.png

86.603

74

72

461418.335

112.5

image84.pngimage84.png

86.603

75

72

467653.718

Through analysis of Table 5, the width still remains the same which is in the same case when the façade was experimented on the shorter side of the base. Also, through using Method 4, the increase in height was found, where for every 1m increase in the height of the roof, height of the cuboid increases by image41.pngimage41.pngm.

Now to find the efficiency when the façade switches sides, we implement the same methodology shown in Method 5.

Fig 7. Area under the curve when the façade switches sides and the curve is at minimum height

image85.png

image86.png

Table 6. Efficiency table when the façade switches sides

Maximum height of roof (m)

Volume of wasted space (m3)

Max volume of cuboid (m3)

Ratio

Volume wasted : Volume cube

75

228230.855

311769.145

0.732 : 1

85

258659.557

353340.443

0.732 : 1

95

28909.494

394905.506

0.732 : 1

110

334740.665

457259.335

0.732 : 1

111

337781.665

461418.335

0.732 : 1

112.5

342346.282

467653.718

0.732 : 1

...read more.

Conclusion

318275.310

Limitations:

Some limitations to this design is of the calculations of the floor area, it does not consider the placements for the stairs and elevators. Also, the designs do not account for thickness of each floor. The sum of each floor may lead to less levels being able to fit if each level must be 2.5m tall.

The efficiency calculations only accounts for the space that is wasted. The other main factor which contributes to this is the financial costs. For example, it may be more financially beneficial by building a single cuboid rather than multiple cuboids due to the amount of materials and the costs of the contractors. Disregarding the aesthetical purposes of the building, it would be more efficient to build a regular rectangular building as it would be able to fit a greater amount of cuboids will less wasted space.

Comparison to Other Structures:

A real life example which follows a parabolic structural design, is the Sydney Harbour Bridge (Fig 9 below).

Fig 9.

image112.png

The Sydney Harbour bridge is 1149m long, 139m tall and 49m in wide. It can be graphed such as in Fig 10 below.

Fig 10. Graph of Sydney Harbour Bridge

image113.png

The design of this building is much different from the one chosen. The one chosen for the building follows the equation, image84.pngimage84.png. The height is 75% of the width, where as the Sydney Harbour Bridge has a height of about 12% of its width.  

...read more.

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