Problem 37 - Calculus: Early Transcendentals

Question

If a surveyor measures differences in elevation when making plans for a highway across a desert, corrections must be made for the curvature of the earth. (a) If \( R \) is the radius of the earth and \( L \) is the length of the highway, show that the correction is \( C = R \sec (L/R) - R \) (b) Use a Taylor polynomial to show that \( C \approx \frac {L^2}{2R} + \frac {5L^4}{24R^3} \) (c) Compare the corrections given by the formulas in part (a) and (b) for a highway that is 100 km long. (Take the radius of the earth to be 6370 km.)

Step-by-Step Solution

Answer

The corrections obtained are very close, validating the approximation.

Step 1: Understanding the Problem

We are given a formula for curvature correction due to Earth's curvature and need to derive it. Additionally, we use a Taylor series expansion to approximate the correction and compare results for a 100 km highway, assuming Earth's radius is 6370 km.

Step 2: Derive the Correction Formula

To derive the correction formula, consider the length of the arc formed by the highway, which is naturally longer than the straight-line distance due to Earth's curvature. By using the trigonometric identity for the secant function, the correction formula is obtained as: \[ C = R \sec\left( \frac{L}{R} \right) - R \]. Here, \( \sec(\theta) = \frac{1}{\cos(\theta)} \) and represents the factor by which the arc exceeds the radius.

Step 3: Using Taylor Series for Approximation

We need the first few terms of the Taylor series expansion of \( \sec(x) \) around \( x = 0 \): \[ \sec(x) = 1 + \frac{x^2}{2} + \frac{5x^4}{24} + \cdots \]. Substitute \( x = \frac{L}{R} \) to approximate \( \sec\left(\frac{L}{R}\right) \). The first two terms yield: \[ \sec\left(\frac{L}{R}\right) = 1 + \frac{L^2}{2R^2} + \frac{5L^4}{24R^4} + \cdots \]. Hence, substituting in the correction formula \( C = R(\sec(\frac{L}{R}) - 1) \) gives: \[ C \approx \frac{L^2}{2R} + \frac{5L^4}{24R^3} \].

Step 4: Calculate and Compare Corrections

First, calculate \( C \) using exact formula for \( L = 100 \) km and \( R = 6370 \) km: \( C = 6370 \sec\left(\frac{100}{6370}\right) - 6370 \). Approximate \( \sec\left(\frac{100}{6370}\right) \) using Taylor series, then plug it into the original formula for exact calculation and the approximate correction: \[ C \approx \frac{100^2}{2 \times 6370} + \frac{5 \times 100^4}{24 \times 6370^3} \].By calculating, we can compare both results.

Key Concepts

Curvature Correction

When planning highways or railways over long distances, surveyors need to account for the curvature of the Earth. This curvature causes the actual distance over a curved surface to be slightly longer than the straight-line distance or map distance.
Curvature correction helps adjust measurements to consider this extra distance:
- Given the Earth's radius (\( R \)) and the length of the highway (\( L \)), the correction is calculated with the formula: \[ C = R \sec\left(\frac{L}{R}\right) - R \]- The secant function (\( \sec \)) accounts for the curve, indicating how much longer the arc Y, across the Earth, is compared to a straight line through its radius.

This concept is crucial in engineering projects, ensuring constructions are accurate and safe across varying terrains.

Taylor Polynomial

A Taylor polynomial is a series expansion that allows us to approximate functions around a specific point. In practical applications, such as curvature correction, Taylor polynomials help simplify complex calculations by approximating functions with polynomials.
For the curvature correction using Taylor Series:
- We need to approximate the secant function \( \sec(x) \) around \( x = 0 \).- The series expansion is: \[ \sec(x) = 1 + \frac{x^2}{2} + \frac{5x^4}{24} + \cdots \]- By substituting \( x = \frac{L}{R} \), we can find the approximation: \[ \sec\left(\frac{L}{R}\right) \approx 1 + \frac{L^2}{2R^2} + \frac{5L^4}{24R^4} \]This provides a simplified equation to calculate the curvature correction practically.The formula becomes much easier to handle, making it feasible for tasks requiring quick estimations.

Trigonometric Identities

Trigonometric identities are equations related to trigonometric functions like sine, cosine, and secant, which provide valuable tools for solving mathematical problems involving angles.
They are crucial in problems related to curvature correction because they allow the transformation and simplification of trigonometric expressions.
In the context of Earth's curvature:- Understanding \( \sec(x) = \frac{1}{\cos(x)} \) helps relate the curvature to angles formed when considering Earth's surface as a part of a large circle.- By applying such identities, surveyors and engineers can accurately transform and compute distances considering curved surfaces.

Trigonometric identities thus enable the simplification of complex problems, ensuring precise adjustments are made in real-world scenarios.

Earth's Radius

Earth's radius is a fundamental parameter used when calculating distances and corrections that consider the planet's curvature.
Generally, for these calculations, a mean radius of about 6370 km is taken, though minor variations may exist depending on geographical location.
- Utilizing Earth's radius (\( R \)) facilitates the conversion of linear measurements (like highways) into arc distances over the Earth's surface.- Accurate knowledge of Earth's radius is essential for deriving correction formulas such as: \[ C = R \sec\left(\frac{L}{R}\right) - R \]This helps in ensuring that the initial measurements are correctly adjusted, leading to precise planning and construction in surveying tasks.
By understanding the dynamic use of Earth's radius in calculations, better strategies and corrections can be applied to infrastructural projects spanning large distances.