Problem 27 - Essential 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 $$C \approx \frac{L^{2}}{2 R}+\frac{5 L^{4}}{24 R^{3}}$$ Compare the corrections given by the formulas in parts (a) and (b) for a highway that is 100 \(\mathrm{km}\) long. (Take the radius of the earth to be 6370 \(\mathrm{km.}\) )

Step-by-Step Solution

Answer

Exact correction is 0.7841 m; approximation is 0.785 m.

Step 1: Understanding the Problem

In this exercise, we need to derive the formula for the correction needed when measuring elevations over a distance due to the Earth's curvature. Part (a) involves deriving an exact expression, and part (b) involves using a Taylor series to approximate the expression.

Step 2: Derive the Exact Correction Formula

Using the geometry of a circle, the correction would be the difference in distance from the arc of the circle (the Earth's surface) and the straight line (the horizontal distance). The horizontal length across the surface is given by \(R \sec(L/R)\), and the straight line is \(R\). Therefore, the correction formula is given by \(C = R \sec(L/R) - R\).

Step 3: Apply Taylor Series Expansion for Approximation

To approximate \(\sec(x)\), use the Taylor series: \(\sec(x) = 1 + \frac{x^2}{2} + \frac{5x^4}{24} + \cdots \).Substitute \(x = L/R\): \(\sec(L/R) \approx 1 + \frac{(L/R)^2}{2} + \frac{5(L/R)^4}{24}\).Substitute this back into the correction formula \(C = R (\sec(L/R) - 1)\) to obtain: \(C \approx R \left(\frac{1}{2} \left(\frac{L}{R}\right)^2 + \frac{5}{24} \left(\frac{L}{R}\right)^4\right) = \frac{L^2}{2R} + \frac{5L^4}{24R^3}\).

Step 4: Calculate Exact Correction for Given Length

Use the formula derived in part (a). For \(L = 100\ \text{km}\) and \(R = 6370\ \text{km}\): \(C = 6370 \cdot \sec(100/6370) - 6370\).First, calculate \(\sec(100/6370)\):\(\sec(0.0157) \approx 1 + \frac{0.0157^2}{2} = 1.000123\).Then:\(C = 6370 \times 1.000123 - 6370 = 0.7841\ \text{m}\).

Step 5: Calculate Approximate Correction Using Taylor

Using the approximation formula from part (b), replace the values:\(C \approx \frac{100^2}{2 \times 6370} + \frac{5 \times 100^4}{24 \times 6370^3}\).Calculate each term:\(\frac{10000}{12740} = 0.785 \ \text{m}\), and\(\frac{5 \times 10^8}{24 \times 258523330000} \approx 0.0001 \ \text{m}\).Therefore, total correction \(C \approx 0.785 + 0.0001 = 0.785\ \text{m}\).

Step 6: Compare Exact and Approximate Corrections

The exact correction using the formula from part (a) is \(0.7841\ \text{m}\), whereas the approximation using the Taylor polynomial is \(0.785\ \text{m}\). These values are very close, demonstrating that the Taylor approximation is accurate for a 100 km highway.

Key Concepts

Curvature of the Earth

The Earth is not flat, which is a principle we often take for granted in math and science. However, when conducting engineering projects such as highway construction, this curvature becomes significant. Imagine a stretch of highway built across a planet-sized sphere. The arc of the Earth's surface follows a circular path, bending subtly over distances.
This bending, or curvature, must be accounted for to ensure accuracy in construction projects.

• Measurements over long distances without considering Earth's curvature can lead to discrepancies in elevation and alignment.
• The curvature correction ensures that plans reflect actual conditions, maintaining the integrity of the engineering design.

Recognizing the Earth's curvature is essential when extrapolating small measurements over larger distances, providing precision in surveying tasks.

Geometric Approximation

Approximating curves and angles in geometric calculations is a common challenge, especially as one deals with more complex shapes like circles or, in this case, spheres. Geometric approximation involves using simpler mathematical expressions to estimate more complex or less intuitive results.
In the problem of Earth's curvature for highway planning, spherical geometry is at play.

• The highway is imagined as a straight line tangent to the Earth's curve, while the actual surface follows a slight arc.
• The approximation involves using geometry to grasp the smaller angle or changes introduced by this curved path.

By understanding and applying these geometric principles, surveyors can make necessary adjustments in their measurements, bridging the gap between calculated models and real-world scenarios.

Taylor Polynomial

The Taylor polynomial is a powerful mathematical tool used to approximate function values. It helps in estimating results that are close to the real ones, especially when dealing with complex functions or series. Using Taylor series expansion, we can break down functions like trigonometric, exponential, or logarithmic functions into polynomial expressions that are easier to use in calculations.
In the exercise at hand, Taylor polynomial is used to approximate the secant function, which forms part of the correction formula:

• The secant function (sec) is approximated through a Taylor series because it's highly useful for small angles.
• This approximation \( \sec(x) \approx 1 + \frac{x^2}{2} + \frac{5x^4}{24} \) helps translate between the geometric ideal and real-world application, narrowing down to practical numbers we can work with.

Through these polynomial approaches, engineers and mathematicians can efficiently solve real-world problems involving curvature, like highway construction, using lighter, simplified calculations.