S L Loney Solutions for Chapter: Heights and Distances, Exercise 1: Examples XXXIII

The elevation of a steeple at a place due south of it is $45°$and at another place due west of the former place, the elevation is $15°$. If the distance between the two places be $‘a’$, prove that the height of the steeple is $\frac{a\left(\sqrt{3}-1\right)}{2\sqrt[4]{3}}.$

A person stands in the diagonal produced of the square base of a church tower, at a distance $2a$ from it, and observes the angles of elevation of each of the two outer corners of the top of the tower to be $30°$, whilst that of the nearest corner is $45°$. Prove that the breadth of the tower is $a\left(\sqrt{10}-\sqrt{2}\right)$.

A person standing at a point $A$ due south of a tower built on a horizontal plane observes the altitude of the tower to be $60°$. He then walks to $B$ due west of $A$ and observes the altitude to be $45°$ and again at $C$ in $AB$ produced he observes it to be $30°$. Prove that $B$ is midway between $A \& C$.

At each end of a horizontal base length of $2a$, it is found that the angular height of a certain peak is $\theta$ and that at the middle point it is $\varphi$. Prove that the vertical height of the peak is $\frac{\alpha \sin \theta \sin \varphi }{\sqrt{\sin \left(\varphi +\theta \right)\sin \left(\varphi -\theta \right)}}$.

$A$ and $B$ are two stations $1000 m.$ Apart $P$ and $Q$ are two station in the same plane as $AB$ and on the same side of it the angles $PAB, PBA,$ $QAB$ and $QBA$ are ${75}^o, {30}^o, {45}^o$ and ${90}^o$ respectively, find how far $P$ is form $Q$ and how far each is form $A$ and $B$.

At a point on a horizontal plane the elevation of the summit of a mountain is found to be $22°{15}^{\text{'}}$and at another point on the plane $1 \mathrm{km}$ farther away in a direct line its elevation is $10°12\text{'}$, find the height of the mountain.

Form the top of a hill the angle of depression of two successive points, $1 \mathrm{km}$ apart, on level ground and in the same vertical plane with the observer are found to be $5^o$ and ${10}^o$, respectively. find the height of the hill and the horizontal distance to the nearest point. (where $\cot 5°=11.4301\& \cot 10°= 5.6714$)$$

A cliff and a tower stand on the same horizontal plane. The height of the cliff is $140 \mathrm{m}$, and the angles of depression of the top and bottom of the tower as seen from the top of the cliff are $40°$ and $80°$ respectively. Find the height of the tower.