   Chapter 11.11, Problem 37E

Chapter
Section
Textbook Problem

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 ≈ L 2 2 R + 5 L 4 24 R 3 (c) Compare the corrections given by the formulas in parts (a) and (b) for a highway that is 100 km long. (Take the radius of the earth to be 6370 km.) To determine

(a)

To show:

CRsecLR-R

Explanation

1) Concept:

2) Calculation:

We have,

From the figure

L is the length of the arc subtended by the angle θ, so

L= Rθ

LR=θ

From figure,

secθ=R+CR

There

To determine

(b)

To show:

C=L22R+5L424R3

To determine

(c)

To compare:

Formulae in part a and b for L=100km and R=6370km

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