IB math HL IA modeling a functional building
- 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.