1. Heejoo Jung
IB MATH HL Per 6
Modeling a functional building
The building will have a rectangular base 150m long and 72m wide, with the
maximum height between 36m and 54m, and the roof structure modeled by parabola.
Step 1: create a model for the curved roof structure when the height is 36m.
When the maximum height of the building is 36m, by thinking as if the side of building is on
axis and assuming that the origin is at the midpoint of the shorter side of the base, the model
of parabolic roof structure has a vertex on the point (0, 36) and the endpoints on (-36,0) and
(36,0).
Thus, the following equation can model the roof structure:
By using the endpoints of the bases above as the parabola’s x-intercepts,
Solving the equation above gives . Therefore, when the maximum height of the
building is 36m, the roof structure can be modeled by the following equation:
Therefore, from the information above, the roof structure will resemble the following graph:

2. Heejoo Jung
IB MATH HL Per 6
▲Figure 1: the graph above represents the parabolic structure of the building. The origin (0,
0) is the midpoint of the shorter side of the base, and the parabolic structure is symmetrical
with respect to the y-axis.
Step 2: Find the dimension of the cuboid with maximum volume which would fit inside this
roof structure.
The dimensions of the cuboid with maximum volume that will fit under the roof structure
above can be calculated with the information from the equation above and that rectangular
base has 150m long and 72m wide. With x representing the distance between the midpoint of
base and endpoints of base, the volume of this cuboid will be:
Expand the equation above and get:
To find the maximum volume,
derive the volume equation and get:
and set

3. Heejoo Jung
IB MATH HL Per 6
(since x represents the distance)
With the value of x, , find the value of y which represents the height of this cuboid:
Since the origin is located on the midpoint of the base, the width of the cuboid with
maximum vlume that fits under the parabolic roof structure with height of 36m is
Therefore, the dimensions of the cuboid under the parabolic roof structure with maximum
volume are (in meters).
The façade of the cuboid will resembles the following graph:
▲Figure 2: the graph above represents the façade of the office cuboid under a parabolic roof
structure with height of 36m.

4. Heejoo Jung
IB MATH HL Per 6
Step 3: Use technology to investigate how changes to the height of the structure affect the
dimensions of the largest possible cuboid.
First, the general equation is needed. To find general equation, let of the roof
structure. Then the vertex of parabola is located in (0, h). Thus the following equation can
model the parabola.
With the x-intercept (36,0), a value of a in terms of h is the following:
Then, the following equation can express the volume of the cuboid:
To determine the maximum volume.
derive the volume equation
and set as 0:
See that h can be cancelled out in both sides, and this shows that the height of the roof
structure does not affect on the width of the cuboid.
Solving the equation above provides x value of .
Thus, whatever the height is, the length of the cuboid that would fit under the parabolic roof
structure with maximum volume is always:
With the information above and Microsoft Excel, how the changes to the height of the
structure affect the dimensions of the largest possible cuboid is investigated.

5. Heejoo Jung
IB MATH HL Per 6
Height of Height of Cuboid Width of Cuboid
Roof(h) (p)
36 24 41.5692194
37 24.66667 41.5692194
38 25.33333 41.5692194
39 26 41.5692194
40 26.66667 41.5692194
41 27.33333 41.5692194
42 28 41.5692194
43 28.66667 41.5692194
44 29.33333 41.5692194
45 30 41.5692194
46 30.66667 41.5692194
47 31.33333 41.5692194
48 32 41.5692194
49 32.66667 41.5692194
50 33.33333 41.5692194
51 34 41.5692194
52 34.66667 41.5692194
53 35.33333 41.5692194
54 36 41.5692194
55 36.66667 41.5692194
56 37.33333 41.5692194
▲Table1: Changes to the height of the structure and the resultant changes in the dimensions
of the cuboid with maximum volume which can fit under the roof structure.
From the Table 1,
The height of the cuboid can be expressed in terms of h by using
.
Thus, the dimensions of a cuboid with maximum volume that lies under the parabolic roof
structure with height h can be expressed as (in meters). This suggests
that the volume of the office block is directly proportional to the height of the roof structure.

6. Heejoo Jung
IB MATH HL Per 6
Step 4: For each height, calculate the ratio of the volume of the wasted space to the volume
of the office block.
By using the equation , set an equation for the volume E, the volume of
the entire building under the parabolic roof structure with height h, is following:
Since 7200h represents the volume of the entire building under the parabolic roof structure
with height of h, the volume of the building is directly proportional to the height of the roof
structure. From the previous finding, it is concluded that the volume of the office space is
also directly proportional to the height of the roof structure. With those two facts, it can be
said that the ratio of the volume of the office space to the volume of the entire building stays
constant whether the height h changes or not.
In addition, because the wasted space W is the following:
When both E and V are proportional to h, the wasted space W is also directly proportional to
h.
Accordingly, it is considered that, regardless of change in height h, the ratio of the volume of
wasted space to the volume of the office space is also constant.
By using Microsoft Excel, the change in volume of the building when height h changes can
be demonstrated. With from previous finding and the equation of the volume of
the cuboid , Excel is used to calculate the ration of the volume of the
wasted space to the volume of the office block with various heights.

7. Heejoo Jung
IB MATH HL Per 6
Height of Height of Width of Volume of Volume of Volume of Ratio
Roof (h) Cuboid(p) Cuboid The Entire Cuboid (V) Wasted (W/V)
Building Space
(E) (W)
36 24 41.5692194 259200 149649.190 109550.810 0.732051
37 24.66667 41.5692194 266400 153806.112 112593.888 0.732051
38 25.33333 41.5692194 273600 157963.034 115636.966 0.732051
39 26 41.5692194 280800 162119.956 118680.044 0.732051
40 26.66667 41.5692194 288000 166276.878 121723.122 0.732051
41 27.33333 41.5692194 295200 170433.799 124766.201 0.732051
42 28 41.5692194 302400 174590.721 127809.279 0.732051
43 28.66667 41.5692194 309600 178747.643 130852.357 0.732051
44 29.33333 41.5692194 316800 182904.565 133895.435 0.732051
45 30 41.5692194 324000 187061.487 136938.513 0.732051
46 30.66667 41.5692194 331200 191218.409 139981.591 0.732051
47 31.33333 41.5692194 338400 195375.331 143024.669 0.732051
48 32 41.5692194 345600 199532.253 146067.747 0.732051
49 32.66667 41.5692194 352800 203689.175 149110.825 0.732051
50 33.33333 41.5692194 360000 207846.097 152153.903 0.732051
51 34 41.5692194 367200 212003.019 155196.981 0.732051
52 34.66667 41.5692194 374400 216159.941 158240.059 0.732051
53 35.33333 41.5692194 381600 220316.863 161283.137 0.732051
54 36 41.5692194 388800 224473.785 164326.215 0.732051
55 36.66667 41.5692194 396000 228630.707 167369.293 0.732051
56 37.33333 41.5692194 403200 232787.629 170412.371 0.732051
▲Table 2: The table above shows the changes to the height of the parabolic roof structure
and resultant changes in the dimensions of the office block, cuboid which can fit inside of the
roof structure when the volume is maximum, volume of the entire building, volume of the
office block, volume of the wasted space and the ratio of the wasted space to the volume of
the office block.
From the table 2, it is shown that the ratio of the volume of the wasted space to the volume of
the office block is constant. Algebraically, using the equation of the volume of the cuboid and
the one of the building, and .
Since , the volume of the cuboid .
The volume of the wasted space is the following:
Thus, the ratio of the volume of the wasted space to the volume of the office block is:

8. Heejoo Jung
IB MATH HL Per 6
Notice that this value equals the one of the ration shown in Table 2.
Step 5: Determine the total maximum office floor area in the block for different values of
height within the given specifications.
Because that the width of the office block at maximum volume is , regardless of change
in height of the roof structure and the length of the building is constant at 150m, the base area
of the cuboid is constant:
It has been supposed that the height of the roof has to be between 36m and 54m:
Since the height of the office block (p) is expressed by the equation ,
From the information given, because the minimum height of a room in a public building is
2.5m and , the maximum number of the floors that will fit inside the office
block with height 24m is 9 floors. The minimum height of the office block to have 10 floors
is .
Thus, the total maximum office floor area when is the following:
.
The minimum height of the office block in this building to have 11 floors is

9. Heejoo Jung
IB MATH HL Per 6
Thus, the maximum number of the floors that will fit inside the office block when
is 10 floors, and the total maximum office floor area is:
.
With the same way, calculations for total maximum office floor area in resultant ranges of
height will be done.
To show the maximum office floor area in the office block with various heights, a table is
made by Microsoft Excel:
Height of Height of Office Maximum Area of Base Total Maximum
Building (h) Block (p) Number of Floor area
Floors
36≤h<37.5 24≤p<25 9 6235.38291 56118.446
37.5≤h<41.25 25≤p<27.5 10 6235.38291 62353.829
41.25≤h<45 27.5≤ p <30 11 6235.38291 68589.212
45≤h<48.75 30≤ p <32.5 12 6235.38291 74824.595
48.75≤h<52.5 32.5≤ p <35 13 6235.38291 81059.978
52.5≤h<54 35≤ p <36 14 6235.38291 87295.361
▲Table 3: Total maximum floor area for various ranges of heights of the building and office
block.
When the façade is one the shorter side of the base, the possible maximum number of floors
is 14 floors if we maximize the volume of the office block.
Step 6: Given that the base remains the same , investigate what would
happen if the façade is placed on the longer side of the base.
To see what would happen if the side of façade is switched, let the façade of the building be
on the longer side of the base 150m.
As in the Step 1, this roof structure can also modeled by parabola, but the origin has to be on
the midpoint of the longer side of the base. Now the range of the height of the roof structure h
is , and the parabolic model of the roof structure will has a vertex on (0,h)
and x-intercept on (-75,0) and (75,0).
Thus, the following equation can model the roof structure:
By plug in the x-intercept (75, 0) into this equation, find the value of a.

10. Heejoo Jung
IB MATH HL Per 6
Graphically, the roof structure will resemble the following:
▲Figure 3: The graph above represents the roof structure with height of 75m and the façade
on the longer side of the base. The midpoint of the longer side of the base is located at the
origin (0,0), and the structure is symmetrical with respect to the y-axis.
The dimensions of the cuboid with maximum volume that will fit under the roof structure
above can be calculated with the information from the equation above and that rectangular
base has 150m long and 72m wide. With x representing the distance between the midpoint of
base and endpoints of base, the volume of this cuboid will be:
Expand this equation and get:
Derive the above equation to find the maximum volume:

11. Heejoo Jung
IB MATH HL Per 6
Set as zero to find the value of x, and get:
Thus, from the given specifications, the length of the cuboid when the volume is maximum,
which fits under the parabolic roof structure, is the following:
(The height of the roof structure does not matter)
The façade of the building will graphically resemble the following:
▲Figure 4: The graph above represents the façade of the building when the height of the
roof structure is 75m.
If is plugged back into the equation ,

12. Heejoo Jung
IB MATH HL Per 6
Thus, the height of the cuboid that can fit under the roof structure when volume is maximum
is . See that the relationship between the height of the roof and the height of the cuboid is
same as when the façade is on the shorter side of base.
Thus, the dimension of a cuboid that can fit under the roof structure with maximum volume
and height h can be expressed as (in meters).
To find the volume of the entire building with the parabolic roof structure with height of h,
use the previous equation , set an equation for the volume E, and get:
Thus, the volume of the entire building with parabolic structure of height h is 7200h, and the
relationship between the volume of the entire building and the height of the roof when the
façade is on the longer side is same as the one when the façade is on the shorter side.
Since it is established that the width of the cuboid with maximum volume, regardless of
change in height of the roof structure, is always , and the length of the building is
constant at 72m, the base area of the cuboid is constant:
Here, see that this base area is same as the one with façade on shorter side of the base.
The base area of the office block and the relationship between the height of the roof and the
height of the office block with façade on longer side of base are the same as with façade on
shorter side of base. However, from the change of the width of the structure to 150m, the
range of the possible heights of the roof structure now is . Then it is

13. Heejoo Jung
IB MATH HL Per 6
suggested that the office space can have a larger number of the floors. Thus, this means that
the volume of the cuboid and the total maximum floor area will increase compared to the case
when the façade is on shorter side.
Height of Height of Length of Volume of Volume of Volume of Ratio
Roof (h) Cuboid (p) Cuboid The Entire Cuboid (V) Wasted (W/V)
Building (E) Space(W)
75 50 86.6025404 540000 311769.145 228230.8546 0.732051
77 51.33333 86.6025404 554400 320082.989 234317.0108 0.732051
79 52.66667 86.6025404 568800 328396.833 240403.1669 0.732051
81 54 86.6025404 583200 336710.677 246489.3230 0.732051
83 55.33333 86.6025404 597600 345024.521 252575.4791 0.732051
85 56.66667 86.6025404 612000 353338.365 258661.6353 0.732051
87 58 86.6025404 626400 361652.209 264747.7914 0.732051
89 59.33333 86.6025404 640800 369966.052 270833.9475 0.732051
91 60.66667 86.6025404 655200 378279.896 276920.1036 0.732051
93 62 86.6025404 669600 386593.740 283006.2598 0.732051
95 63.33333 86.6025404 684000 394907.584 289092.4159 0.732051
97 64.66667 86.6025404 698400 403221.428 295178.5720 0.732051
99 66 86.6025404 712800 411535.272 301264.7281 0.732051
101 67.33333 86.6025404 727200 419849.116 307350.8842 0.732051
103 68.66667 86.6025404 741600 428162.960 313437.0404 0.732051
105 70 86.6025404 756000 436476.804 319523.1965 0.732051
107 71.33333 86.6025404 770400 444790.647 325609.3526 0.732051
109 72.66667 86.6025404 784800 453104.491 331695.5087 0.732051
112.5 75 86.6025404 810000 467653.718 342346.2820 0.732051
▲Table 4: Changes in the dimensions of the office block, volume of the entire building
under the roof, volume of the office block, volume of the wasted space, and the ratio between
the wasted space and the volume of the office block, as the height of the parabolic roof
structure changes.
Notice that the ration between the wasted space and the volume of the office block is the
same as when the façade is on shorter side, and the volume of the entire building under the
roof structure increased (comparing with the Table 2)

14. Heejoo Jung
IB MATH HL Per 6
Height of Height of Office Maximum Area of Base Total Maximum
Building (h) Block (p) Number of Floor area
Floors
75≤h<78.75 50≤c<52.5 20 6235.382907 124707.6581
78.75≤h<82.5 52.5≤c<55 21 6235.382907 130943.0411
82.5≤h<86.25 55≤c<57.5 22 6235.382907 137178.4240
86.25≤h<90 57.5≤c<60 23 6235.382907 143413.8069
90≤h<93.75 60≤c<62.5 24 6235.382907 149649.1898
93.75≤h<97.5 62.5≤c<65 25 6235.382907 155884.5727
97.5≤h<101.25 65≤c<67.5 26 6235.382907 162119.9556
101.25≤h<105 67.5≤c<70 27 6235.382907 168355.3385
105≤h<108.75 70≤c<72.5 28 6235.382907 174590.7214
108.75≤h<112.5 72.5≤c<75 29 6235.382907 180826.1043
h=112.5 c=75 30 6235.382907 187061.4872
▲Table 5: Total maximum floor area in various ranges of heights of the building.
Comparing this table, Table 5, with the Table 3, it is found that the total maximum area
significantly, as the height of the cuboid and the maximum number of floors increased.
Step 7: You now decide to maximize office space even further by not having the block in the
shape of a single cuboid.
The Figure 2 and 4 shows that large portion of the space under the roof structure is wasted
with the office space designed in a single cuboid. As shown in the Table 2 and 4, the ratio of
the volume of the wasted space to the volume of the office block in both case, either the
façade on shorter side or longer side, is nearly 0.732, which suggests that the 73.2 % of the
office space is wasted.

15. Heejoo Jung
IB MATH HL Per 6
If office space under this roof structure is maximized even further, the building will resemble
the following:
▲Figure 5: The graph above represents the façade of the building not having the block in
shape of a single cuboid with the midpoint of the shorter side of the base on origin and the
height of the roof structure as 36m. As shown in the graph above the layers of cuboids have
different dimensions. Comparing this figure to Figure 2 and 4, it is shown that this design
provides much larger office space and much less wasted space.
Here, the reason why the graph of inverse of the original equation is that original equation
makes more work to do, since the rectangles are created vertically, not horizontally, and it is
better to use the inverse of the equation rather than to try to make horizontal rectangles. Thus,

16. Heejoo Jung
IB MATH HL Per 6
actually, the graph resembles half of the façade of the building that is rotated to clock
wise.
Notice that the design with cuboids with various dimensions provides a larger number of
floors than the designs with a single cuboid does. While the designs with single cuboid
provide only 9 floors when the height is 36m, the design with cuboids with various cuboids
provides 14 floors with the same height, 36m.
With this design which has cuboids with different dimensions, the only thing varies from
floor to floor is width, and the height and the length of office space are constant for every
floor.
As it is in previous case, the roof structure is modeled by the following general equation:
(h represents the height of the roof structure)
From the equation, it is expected that the distance between the midpoint of the base and the
endpoint of the each floor (x) will be related to the height of the building that each floor
reaches (y).
The minimum height of each floor, which is 2.5m, will be used for every floor, since the
design has to maximize the number of floors.
By solving the equation
in terms of x, and get:
Since the width is 2x, it depends on what number of floor it is and how long the height of the
roof structure is. In this case, since y represents the height of the floor that each floor reaches,
y will be a multiple of 2.5(minimum height of the floor), which can express y as 2.5a.
Let n be the number of the floors that fits under the parabolic roof of the structure with height
h. The length of the floor space is 150m and the height of the space is 2.5m for every floor.
Then the equation for the volume of the office space is:

17. Heejoo Jung
IB MATH HL Per 6
From the equation , can be expressed as the following:
To get the equation for the total volume of the office space that fits under the parabolic roof
structure with height of h, combine those two equations and get:
The total office floor area, , is:
Notice that both the maximum number of floors and the height of the roof structure also have
effect on the total office floor area.

18. Heejoo Jung
IB MATH HL Per 6
With h having effect on n and f and n having effect on volume, the volume equations for
different ranges can be supposed. Remember that the height h is between 36 and 54, and that
the number of floor n is found by dividing height h by 2.5.
Height of Maximum Volume Equation
Building(h) number
of Floors
(n)
36≤h<37.5 14
37.5≤h<40 15
40≤h<42.5 16
42.5≤h<45 17
45≤h<47.5 18
47.5≤h<50 19
50≤h<52.5 20
52.5≤h<54 21
▲Table 6: Volume equation with different ranges of height of the roof structure.

19. Heejoo Jung
IB MATH HL Per 6
Step 8: Review your model and calculate the increase in floor area and the new volume ratio
of wasted space to office block.
As shown in the Table 2 and Table 3, the volume of the office space is maximized when the
height of the roof structure is maximized. Similarly, the total area is also maximized when the
height of the roof structure is maximized, because . Thus, the case of possible
maximum height h, which is 54m, can be used to compare the maximum volume of office
space and maximum office floor area when the building contains a single cuboid with those
when the building contains cuboids with various dimensions.
Microsoft Excel will be used to calculate the change between the two cases.
With a Single With Cuboids of Change
Cuboid Different Dimensions
Volume of Entire Building 388800 388800 0
Maximum number of Floors (n) 14 21 7
Maximum Total Office Floor area 87295.36 150198.98 62903.6208
Maximum Volume of Office Block 224473.8 375497.455 151023.67
Volume of Wasted Space (w) 164326.2 13302.546 -151023.67
Ratio (W/V) 0.732051 0.0354265 -0.69662435
▲Table 7: The table above shows the comparison between the two cases, with a single
cuboid and with cuboids of different dimensions under the parabolic roof structure when the
height of the roof structure is 54m, which is the highest possible height.
The Table 7 shows that the maximum total office floor area and maximum volume of office
block significantly increase when the building contains the different sized cuboids, since the
increase in maximum total office floor are is nearly 62903 and the one of maximum
volume of office block is approximately 151024 . In addition, notice that the decrease of
the ratio to the volume of wasted space to the volume of office block is 0.697, and the wasted
space with different sized cuboids is only 3.54% of the volume of office space.
In conclusion, the building containing cuboids with different dimensions is preferable to the
one with a single cuboids, since the one with various cuboid allows more efficient use of
space for the parabolic roof structure. Additionally, 54m is recommended for the height of the
structure that maximizes the volume of office space and the total office floor area.