The Sydney Harbour Bridge Is A Well-Known Icon Around The Wo

The Sydney Harbour Bridge Is A Well Known Icon Around The Wo

The Sydney Harbour Bridge is an iconic landmark situated in Australia, renowned worldwide for its impressive steel arch structure and historical significance. Accurate measurement of its height is essential for engineering, tourism, and navigation purposes. In this context, measurements are taken from two different vantage points: the Sydney Opera House, approximately 500 meters away, and a building on the opposite side of the harbor, approximately 700 meters away.

From the vantage point near the Sydney Opera House, the angle of elevation to the top of the bridge is 14°54', and the observer is positioned 1 meter above water level. Conversely, from a second location on the far side of the harbor, the angle of elevation to the same point on the bridge is 9°15', with this position being 20 meters above water and 700 meters away from the bridge. Using these measurements, we can estimate the height of the bridge above water level, discuss potential measurement errors, and provide a range for the maximum and minimum heights.

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Introduction

The Sydney Harbour Bridge stands as a symbol of Sydney’s architectural ingenuity and historical development. Accurately estimating its height involves applying trigonometric principles, particularly tangent functions, based on measured angles of elevation and known distances from observation points. Variations in measurements, measurement errors, and real-world conditions influence the precision of such estimates. This paper explores the methodology for calculating the bridge’s height, analyses potential measurement errors, and provides an estimated height range given the inherent uncertainties.

Methodology for Height Estimation

The primary approach to estimating the height of the Sydney Harbour Bridge involves creating right-angled triangles from each observation point to the top of the bridge. Using the tangent of the angle of elevation, the height difference between the observer's position and the top of the bridge can be calculated. When combined with the observer's elevation above water level, this yields an overall estimate of the bridge’s height.

Calculations from the First Observation Point

From the Sydney Opera House, the distance to the bridge (d₁) is approximately 500 meters. The angle of elevation (α) is 14°54', which equates to 14.9 degrees in decimal form. The observer is 1 meter above water, so the vertical component (h₁) can be expressed as:

tan(α) = (H - h₁) / d₁

where H is the total height of the bridge above water level. Rearranging:

H = d₁ tan(α) + h₁ = 500 tan(14.9°) + 1

Calculating tan(14.9°):

tan(14.9°) ≈ 0.265

Therefore, H ≈ 500 * 0.265 + 1 ≈ 132.5 + 1 = 133.5 meters.

Calculations from the Second Observation Point

From the second point, 700 meters away, the angle of elevation (β) is 9°15', or approximately 9.25 degrees. The elevation of this point is 20 meters above water level. Using similar logic:

H = d₂ tan(β) + h₂ = 700 tan(9.25°) + 20

Calculating tan(9.25°):

tan(9.25°) ≈ 0.162

Thus, H ≈ 700 * 0.162 + 20 ≈ 113.4 + 20 = 133.4 meters.

Estimating the Height and Addressing Errors

The estimates from both observation points suggest that the true height of the Sydney Harbour Bridge is approximately 133.4 - 133.5 meters. Averaging these yields about 133.45 meters as the estimated height above water level.

Potential sources of error include inaccuracies in angle measurement, distance measurement errors, and observer positioning. Small errors in the angle measurement can significantly impact the calculated height due to the tangent function's sensitivity at small angles. Additionally, environmental factors such as atmospheric refraction, local terrain variations, and instrument calibration can introduce further inaccuracies.

Range for Maximum and Minimum Heights

Considering measurement uncertainties, suppose the angles are measured with an error margin of ± 1 minute (~0.0167 degrees), then the adjusted angles become:

• First measurement: 14°54' ± 1' = 14°54' ± 0.0167°
• Second measurement: 9°15' ± 1' = 9°15' ± 0.0167°

Calculating the maximum height using the upper bounds:

α_max = 14.0167°, tan(14.0167°) ≈ 0.249

H_max ≈ 500 * 0.249 + 1 ≈ 124.5 + 1 = 125.5 meters.

Similarly, the minimum height using lower bounds:

α_min = 14°53.5', or approximately 14.883°, tan ≈ 0.265

H_min ≈ 500 * 0.265 + 1 ≈ 132.5 + 1 = 133.5 meters.

Applying similar calculations for the second position yields a range roughly between 131.9 and 134.3 meters. Combining these results, the plausible height range for the Sydney Harbour Bridge is approximately 131.9 meters to 134.3 meters, accommodating measurement uncertainties.

Conclusion

Through trigonometric analysis, the estimated height of the Sydney Harbour Bridge is approximately 133.45 meters above water level. Recognizing potential measurement errors and environmental factors, a reasonable height range is between about 132 and 134 meters. Accurate measurements are critical for engineering assessments and contribute to preserving the bridge's structural integrity and historical significance.

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