Clifton Suspension Bridge Lab Report

Abstract:

In this lab report, we aim to determine the mathematical equation that describes the parabolic curve of the Clifton Suspension Bridge. To achieve this, we initially calculated the vertex coordinates, followed by an investigation into a more accurate equation using Desmos and Microsoft Excel.

The report discusses the methods, calculations, and results obtained in this study.

Introduction

The Clifton Suspension Bridge, a renowned engineering marvel, spans 214.05 meters in length with tower heights of 26 meters. The objective of this study is to mathematically describe the parabolic curve formed by the bridge and calculate its equation.

The lowest point of the parabolic curve above the road is 2 meters.

Methodology

To calculate the vertex (h, k) of the parabolic curve, we use the formula:

x = 214.05 meters / 2 = 107.025 meters

y = 2 meters

Vertex = (107.025, 2)

h = 107.025

k = 2

Now, we can calculate the equation of the parabola using the formula ^y = a(x - h)² + k:

y - k = a(x - h)²

26 - 2 = a(0 - 107.025)²

24 = a(11454.35063)

a = 24 / 11454.35063 = 0.00209527373

Now, we can substitute the values of h, k, and a back into the equation:

y = 0.00209527373(x - 107.025)² + 2

Next, we validate this equation by substituting another point from the parabola, such as (214.05, 26):

y = 0.00209527373(214.05 - 107.025)² + 2

y = 25.99999996 meters

Discussion

The formula for the turning point of a quadratic has provided an estimate of the equation of the Clifton Suspension Bridge's parabolic curve. However, this method has limitations, as it relies on only two points and may not be as accurate with additional refined data points. To improve accuracy, we utilized software tools like Desmos and Microsoft Excel.

In Desmos, a to-scale side-view image of the Clifton Suspension Bridge was utilized. Twelve points, including the vertex and y-intercept, were added to the parabola using the scale. The coordinates of these points were recorded in a table and imported into Microsoft Excel to derive the equation of the parabola. The resulting equation was then brought back into Desmos to create a parabolic line that closely traced the real arch of the Clifton Suspension Bridge. The equation obtained through this process was: y = 0.0325x² - 0.3902x + 1.3738.

Comparing this new equation to the previous, less accurate one, y = 0.00209527373(x - 107.025)² + 2, we can observe that the new equation is in the form of y = ax² + bx + c.

To assess the exactness of the equation, we calculated the coefficient of determination, R², where R² = 0.9954 for the equation derived from Desmos and Excel. This high R² value demonstrates the accuracy and close approximation of this equation to the actual parabolic curve of the Clifton Suspension Bridge.

Conclusion

In conclusion, this study successfully determined the mathematical equation that describes the parabolic curve of the Clifton Suspension Bridge. The use of Desmos and Microsoft Excel allowed for a more accurate approximation, resulting in the equation y = 0.0325x² - 0.3902x + 1.3738. The coefficient of determination, R², of 0.9954 validates the accuracy of this equation. Further refinements and data points may lead to even more precise equations in future studies.