St. Louis Arch Quadratic Equation

DERIVATION OF ST. LOUIS GATEWAY ARCH QUADRATIC EQUATION Submitted by: STUDENT 1 Submitted to: Instructor January 7, 2020 Introduction The famous St. Louis arch are also known as Gateway Arch in Missouri is considered as the world’s tallest arch and also the world’s tallest man-made monument in the Western Hemisphere. Moreover, it is also considered as the states’ tallest accessible building. It is located on the west bank of the Mississippi river and has been a very iconic tourist spot within the united states. This particular arch was chosen by the student for this report because of two reasons, first, it is the tallest arch which somewhat caught his attention, and second, it is very much applicable for the conceptual report as it is required to determine a particular equation for an arch. The objective of this report is to derive the equation of the formed by the arches of the Gateway Arch. Report Development Plan In order to have a well-organized process for the whole report, a development plan has been created made as shown in Figure 1, which became framework from start to finish. The whole work was divided 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 Preliminary Ideas Scheduling Research of Related Information Research for the Arch Dimensions Formulation of Equation Material Identification Budgeting and Buying of Materials Creation of Model Phase 1. Conceptualization To initialize the work, the student has provided an ample time just after the discussion of the work load so that he may be able to write down all his ideas regarding the topic and create a very useful workflow which needs to be followed all throughout the activity. Brainstorming In this part, the student has listed all his preliminary idea of the topic. He has thought of the possible structures which he knows he can use for the report. He also initially planned on how to model the structure and what shall be the material to be used. Also, he tried on how he shall do the derivation of the quadratic equation for the structure. All of his initial ideas are modified and enhanced through the research results presented in the next phase Scheduling To be able to finish the whole task with the given time frame of 1 week, the student 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 student started to gather information that are beyond or might give clarification to his 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 Arch Dimensions During the research, it has been encountered that St. Louis Gateway Arch is a very good example of a parabolic arch which can be used by students in order to demonstrate the real-life application of mathematical equations. Thus, there are numerous sites that has provided a complete dimension of the arch but the student has chosen the much more ideal graphical representation of the St. Louis Gateway Arch shown in the figure below. The dimensions shown in the figure are then tabulated to have clearer values. Dimension Length (Outer to Outer of Base) Length (Center to Center of Base) Height (Top of Arch Vertex) Height (Center of Arch Vertex) Measurement 630 ft 598 ft 630 ft 625 ft In the given table, there are two possible equations that can be created. The first one is the equation for the outermost curve of the arch while the second one is the centerline curve of the arch. For this report, the student has utilized both in order to come up with two different equations. Phase 3. Mathematical Analysis Since it is evident that the curve equation that is suited for the St. Louis Gateway Arch is a quadratic equation, that has been adapted by the student for the 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 vertex form was chosen. The vertex form of quadratic equation is given by the following formula. y=a (x−h )2 +k To get everything started, the three known points was identified within the curve in which the first point is middle portion of the base and was set as the origin. Thus, the arch was symmetrically divided into two, coming up with the points given in the table below. Point A V C Outer Curve x-coordinate y-coordinate -315 0 0 630 315 0 Centerline Curve Point A V C x-coordinate -299 0 299 y-coordinate 0 625 0 Through these, the vertex of the parabola can already be identified which is basically the topmost middle part of the arch which is at coordinate VT (0, 630), giving the values of h and k as 0 and 630, respectively for the external arch, and VB (0, 625), and giving the values of h and k as 0 and 625, respectively for the centerline arch. Then choosing a point A or C among the three given points, the student was able to come up with the value of a and also the whole equation for each arch. The computation is given by the following. For the External Curve y=a (x−h )2 +k 0=a(−315−0 )2 +630 a=−0.00 63492 y=−0 0.0063492 x 2+ 630 For the Centerline Curve y=a (x−h )2 +k 0=a(−299−0 )2 +625 a=−0.00 69910 2 y=−0.0069910 x + 625 Phase 4. Modelling For the model, the student has scaled the real dimensions in order to have smaller dimensions that can fit the maximum required measurements of the model. In order to have a better scale, the dimensions were converted from ft to meters. The scale used is 10 m to 1 cm giving the following dimensions of the model. Dimension Length (Outer to Outer of Base) Length (Center to Center of Base) Height (Top of Arch Vertex) Height (Center of Arch Vertex) Actual (ft) 630 ft 598 ft 630 ft 625 ft Converted (m) 192.027 182.27 192.024 190.5 Model (cm) 19.2 18.2 19.2 19.05 For the materials, the student has utilized black cardboard for the arch, then green and blue cardboard which will represent the surrounding environment. 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. The blue cardboards were then cut out and pasted to the drawn arch. Other objects were used in order to represent cars and buildings in order to make the building more presentable as a model of the St. Louis Gateway Arch. Conclusion In this task, the student was able to derive his quadratic equation of the curvature of St. Louis Gateway Arch and was also able to create his own model, through the ideas that he has learned during the discussion and also through the student’s 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 student because he was able to apply the mathematical discussions and theories into real life. Arches are very much complex structures that are solved through complex solutions and equations, but through this work, the student was able to use his own knowledge in order to create an arch model which they think before as very much beyond his 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 student because everything is exciting. Bibliography “St. Louis Gateway Arch.” (n.d.). Retrieved from