Additional problem related to material about the time of worksheet 2.
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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. Suppose a surveyor, located at the point $P$ in
the figure below, looks horizontally through his scope at a (vertical) 
post held by an assistant located a the point $Q$, with exactly the
same elevation as $P$. The post has markings on it indicating the
height on the post. In this example, $P$ and $Q$ having the same
elevation means that they have the same distance to the center of the
earth.  Then the surveyor
will not see the bottom of the post, but instead observes a height $C$
on the post. If the assistant were standing at a point of lower or
higher elevation, then the surveyor must subtract $C$ from any reading
he obtains on the post.
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\psfrag{L}{$L$}
\psfrag{R}{$R$}
\psfrag{C}{$C$}
\psfrag{P}{$P$}
\psfrag{Q}{$Q$}
\psfrag{pole}{post}

\centerline{\includegraphics[height=1.25in]{highway.eps}}
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\itemitem{(a)} If $R$ is the radius of the Earth and $L$ is the length
of the highway, show that the correction is
$$C=R \sec(L/R) - R.$$
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\itemitem{(b)} Use a Taylor polynomial to show that 
$$C\approx {{L^2}\over{2R}}+ {{5L^4}\over{24R^3}}.$$
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\itemitem{(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.)

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