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.

To find the dimensions of a cuboid with maximum volume under the curve, we find the area of the face of the cuboid:

Let

 

                {sub in  value}

         

 

Differentiate to find maximum area at ,

 

  

 

   

 

             

  

 

            {these values are the roots}

Find the second derivative to prove that this is a maximum curve:

  

 

∴  The function concaves downwards when  

Sub in  value (from the first derivative) into original function to find  value

 

     
 

∴  Dimensions of the maximum volume of the cuboid are:

The width is found by finding the difference between the 2 points as the line of symmetry is designed at the  axis (refer to Fig 2 below).

               

m

 m

 m

And the volume of the cuboid:

∴  These dimensions produce the cuboid with the maximum volume under the graph (also shown in Fig 2 below).

Fig 3. Dimensions of cuboid with maximum volume when  (front view)

Varying Height of Roof:

After determining the dimensions of the maximum cuboid, we test to see how our independent variable (the maximum height of the roof) affects our dependant variable (the cuboid).

Using the same methodology in Method 3, the following table was constructed and calculated using the GDC.

The maximum and minimum height of the roof was chosen. 2 consecutive heights near the maximum where then chose to see the relationship between a 1 meter increases. And the medium height of the roof was chosen.

Table 1. How height of roof affects maximum cuboid dimensions and volume

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 m. For this type of model function, the height of the cuboid can be found by ; where  is the height of the cuboid and  is the maximum height of the roof.

Method 4. Finding the increase in height

Let

Using values of roof height at 52m and 45m

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 .

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, .

Method 5:

To find volume under the curve algebraically, we integrate:

Using GDC to find the answer:

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  axis)

Multiply by length to find the volume:

 

 

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

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:

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

∴   The area of each floor is:

The number of levels is found by:

 ; where  is the height of the office block

∴   The total floor area can be found by:

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

At m

                                     

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

Using Method 7, the following table was constructed.

Table 4. Maximum area of cuboid floor:

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:

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

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

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 m.

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

Table 6. Efficiency table when the façade switches sides

The efficiency is seen to be the same as when the façade was on the shorter side of the base.

After determining the efficiency, we find the area of the cuboid floor once again.

Table 7. Area of cuboid floor when the façade switches sides

Comparing the side of the Façade:

Through analysis of both situations, the façade at the longer length of the base will be chosen to be the design at a maximum height where the roof is 112.5m tall. This due to the fact that provides a larger volume and it provides a larger area floor area.

Increasing the efficiency:

To improve the efficiency, instead of having a single cuboid under the curve, we will have multiple cuboids with each at a fixed height of 2.5m (refer to Fig 8 below)

Fig 8. Multiple cuboids

Method 8. Finding the volume of the multiple cuboids:

To find the volume of the multiple cuboids:

Let

e.g. At level 1:

At level 2:

At level 3:

And etc.

Maximum height

 

The difference between these 2 points is the width

Therefore, multiply the positive root by 2 to obtain the width

The volume:

This can be further generalised by:

                     Where  number of floors

There are 45 floors when the maximum height of the roof is 112.5m.

∴   The equation to finding the volume of multiple cuboids is

Method 9. Using Excel:

To now further validate this statement through the use of Microsoft Excel to construct the follow table below. Column C was found by applying the equation, , to Column C, where  is the value directly left in Column B. This rule was extended until the floor level 45.

Column F was obtained by multiplying the cells in Column C, Column D and Column E in the same row. This rule was also extended down until level 45.

By using the ‘AutoSum’ tool in Excel, the total volume was found and supports the equation above.

Column G was found by multiplying the width and length of each row and once again, the rule was extended until level 45. The ‘AutoSum’ tool was used once again to determine the total floor area.

Table 8. Dimensions and volume of multiple cuboids:

Now, after determining the total volume of the multiple cuboids, we must determine the efficiency by calculating the ratio of wasted space volume to the volume of the cuboids. By implementing Method 5, the ratio was found to be:

This value tells us that the wasted space volume is equivalent to 1.79% of the volume of the multiple cuboids. This is considered to be fairly efficient compared with only have a single cuboid with 73.2%.

The floor is also more efficiently used as a single cuboid has a floor area of 280592.231m2, where as multiple cuboids has a total floor area of 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.

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

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