Taylor Series Applications: Need a proof of this problem

[jbsmith11]

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.

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

 

[tkhunny]

Well, do it.

You'll need a planet with a road on it. You'll also need the same roadm but tangent to the planet.

What's your plan?

 

[galactus]

Just draw a triangle. Label the long side (hypoteneuse) R+C and the radius of the Earth side R. The small side would be L, the highway. The highway is very small compared to the whole Earth. Let x be the central angle.

\(\displaystyle cos(x)=\frac{R}{C+R}\)

\(\displaystyle C+R=\frac{R}{cos(x)}=Rsec(x)\)


For the Taylor series, sub in L/R into the series for sec(x) and continue.