A man $M$ is at the top of a mountain which is of height $h$ km. The radius of the earth is assu

Question

A man $M$ is at the top of a mountain which is of height $h$ km. The radius of the earth is assumed to be a constant $R$ km. The furthest point on the earth’s surface that the man can see is a point $P$ such that $MP=x$ km and the angle $POM=\theta $, where $O$ is the centre of the earth (see diagram). You may assume that the height of the man is negligible. Show that $x={{\left( 2hR \right)}^{\frac{1}{2}}}{{\left( 1+\frac{h}{2R} \right)}^{\frac{1}{2}}}$.[3] It is given that $h$ is small compared to $R$. Show that, if $\alpha =\frac{h}{R}$, $\sin \theta \approx {{\left( 2a \right)}^{\frac{1}{2}}}\left( 1-\frac{3}{4}\alpha \right)$.[5] The man $M$ has a scientific instrument which enables him to estimate the angle between $PM$ and the horizontal. Given that this angle is $2{}^\circ $ and that the radius of the earth is $6375$ km, find estimates for the values of $\alpha $ and $h$. [4]

Accepted Answer

(i) $x=(2hR)^{1/2}\left(1+\frac{h}{2R}\right)^{1/2}$. (ii) $\sin\theta\approx(2\alpha)^{1/2}\left(1-\frac34\alpha\right)$. (iii) $\alpha\approx6.10\times10^{-4}$ ($3$ s.f.) and $h\approx3.89\text{ km}$ ($3$ s.f.).

2017 Specimen Paper P1 Q9

2017 Specimen Paper P1 Q9

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