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DERIVAT ION OF SYDNEY HARBOU R BRIDGE QUADRA TIC EQUATI ON

Submitted by: STUDENT 1 STUDENT 2

Submitted to: Instructor

January 6, 2020 Introduction Sydney Harbour Bridge in Sydney NSW, Australia is a very iconic bridge since it is the largest steel arch bridge in the world. It was constructed in a period of 8 years by using steel as a material, with a tallied weight of 53,000 tonnes, with the additional 6 million hand-driven rivets. It was opened to the public in 1932 and was used by vehicles for transport in the two sides of the bridge. Since then, it became famous to the people as it gives a stunning view as people climb on top of it. This particular bridge was chosen by the group for their report because of two reasons, first, it is the largest arch bridge, and second, it is very much applicable for the conceptual report as it is required to determine a particular equation for a bridge. The objective of this report is to derive the equation of the formed by the arches of the Sydney Harbour Bridge.

Report Development Plan In order to have a well-organized task for the whole report, the group has made their own report development plan shown in Figure 1, which became their framework from start to finish. The group has divided the whole work into four different phases namely the conceptualization part, data gathering, mathematical analysis, and modelling. Each phase is discussed in the succeeding sections together with the inputs included within that phase.

1. Conceptualization

2. Data Gathering

3. Mathematical Analysis

4. Modelling

Brainstorming Division of Work Scheduling

Research of Related Information Research for the Bridge Dimensions

Formulation of Equation

Material Identification Budgeting and Buying of Materials Creation of Model

Phase 1. Conceptualization To initialize the work, the group has gathered for a short meeting in order to know the ideas of each member which can be useful for the completion of work. The group also has designated the tasks for each member and was able to develop a schedule which is aligned with the allocated time interval for the completion of the report.

Brainstorming In this part, each member has given their personal ideas or prior knowledge regarding the topic most specifically on the critical part of the report which is the application of quadratic equation in real life. Although discussions during the course are mostly theoretical, it did not hinder the group on applying it to bridge since its shape is known to be parabolic which can easily be drawn and graphed as the group usually does in their calculations. Division of Work Since the group contains two members, it has been decided that each member will have to take half for the data gathering, one for the researching of related information and the other member is for the research of bridge dimensions. Then, the two members will do the analysis and modeling together. Scheduling

To be able to finish the whole task with the given time frame of 1 week, the group has developed a schedule in which the deadlines in this schedule should be strictly met. Day 1

Day 2 Day 3 Day 4 Day 5

Day 6

Preliminary Meeting

Research Period Compiling of Research Results Mathematical Analysis Buying of Materials

Modelling

Phase 2. Data Gathering This phase of the whole process is the part wherein the group started to gather information that are way beyond their knowledge or might give clarification to their prior knowledge. This process is vital for the completion of work because the input information necessary for mathematical analysis must be gathered first with sufficient evidence. Research of Related Information The ideas that were gathered in this part were used in order to give a better presentation of the work and most of them are already presented in the introduction. Research of Bridge Dimensions During the research, it has been encountered that Sydney Harbour Bridge is a very good example of a parabolic arch which has been used by many schools in order to demonstrate the real-life application of mathematical equations. Thus, there are numerous sites that has provided a complete dimension of the bridge but the group has chosen the official site of the Sydney Harbour Bridge. The dimensions gathered by the group are presented in the given table below. It must be noted that the following measurements are from mean sea level thus might vary depending on the time and day it was measured.

Dimension

Measurement

Total Length

1,149 m

Utilized for Calculation? No

Width Height (Top of Truss) Height (Bottom of Truss) Longest Span

48 m 141 m 134 m 503 m

No Yes Yes Yes

In the given table, the two dimensions necessary for the calculation are the height of the bridge, which will serve as the height of the curve, and the longest span which will serve as the base of the curve. The total length pertained to this table is the arc length of the curve, thus, should not be interchanged with the description of the longest span.

Phase 3. Mathematical Analysis Since it is evident that the curve equation that is suited for the Sydney Harbour Bridge is a quadratic equation, that has been adapted by the group for their derivation. In this phase, the equation for the parabola is derived using the inputs gathered in the phase 2. Within three types of quadratic equation namely, intercept, vertex, and standard, the group has chosen the vertex form which is usually used for parabolas. The vertex form of quadratic equation is given by the following formula. y=a (x−h )2 +k To get everything started, the group has identified three known points within the curve in which the first point is middle portion of the base as their origin. Thus, the bridge was symmetrically divided into two coming up with the points given in the table below. Top Truss Points Point A V C Point A V C

x-coordinate y-coordinate -251.5 0 0 141 251.5 0 Bottom Truss Points x-coordinate y-coordinate -251.5 0 0 134 251.5 0

Through these, the vertex of the parabola can already be identified which is basically the topmost middle part of the bridge truss which is at coordinate VT (0, 141), giving the values of h and k as 0 and 141, respectively for the top truss, and VB (0, 134), and giving the values of h and k as 0 and 134, respectively for the bottom truss.

Then choosing a point A or C among the three given points, we were able to come up with the value of a and also the whole equation for each trusses. The computation is given by the following. For the Top Truss y=a (x−h )2 +k 0=a(−251.5−0 )2+ 141 a=−0.0022291697 2

y=−0.0022291697 x +141

For the Bottom Truss y=a (x−h )2 +k 2

0=a(−251.5−0 ) + 134 a=−0.0021185017 2

y=−0.0021185017 x +134

Phase 4. Modelling For the model, the group has scaled the real dimensions in order to have smaller dimensions that can fit the maximum required measurements of the model. The scale used is 10 m to 1 cm giving the following dimensions of the model. Dimension Height (Top of Truss) Height (Bottom of Truss) Longest Span

Real (m) 141 m 134 m 503 m

Model (cm) 14.1 cm 13.4 cm 50.3 cm

For the materials, the group has utilized black cardboard for the bridge and blue cardboard which will represent the sea. The measurements were traced into the cardboards to have an accurate model. These were then designed and painted in order to have a presentable

appearance. Styrofoam was also used as supplementary design material which was used as the base of the sea, and the blue cardboards were then cut out and pasted to the drawn bridge. Other objects were used in order to represent cars and buildings in order to make the building more presentable as a model of the Sydney Harbour Bridge.

Conclusion In this task, the students were able to derive their quadratic equation of the curvature of Sydney Harbour Bridge and was also able to create their own model, through the ideas that they have learned during the discussion and also through their hard work on researches. The final outputs have been successful although with minimal errors which might have caused by some common sources of errors.

Reflection This report has widened the mind of the students because they were able to apply the mathematical discussions and theories into real life. Bridges are very much complex structures that are solved through complex solutions and equations, but through this work, the group was able to use their own knowledge in order to create a bridge model which they think before as very much beyond their capacities. This student work is very much helpful for the students in order to appreciate mathematics. Finishing each part of the task has been an achievement for the group because everything is exciting.

Bibliography “Info about the Sydney Harbour Bridge.” Info about the Sydney Harbour Bridge., www.sydneyharbourbridge.info/. “Facts and History of Sydney Harbour Bridge.” Sydney Harbour Bridge - Facts, History and Interesting Information, www.bridgesdb.com/bridge-list/sydney-harbour-bridge/....