M L Aggarwal Solutions for Chapter: Heights and Distances, Exercise 3: Chapter Test

There is a small island in between a river $100 \mathrm{m}$ wide. A tall tree stands on the island. $\mathrm{P}$ and $\mathrm{Q}$ are points directly opposite each other on the two banks, and in line with the tree. If the angles of elevation of the top of the tree from $\mathrm{P}$ and $\mathrm{Q}$ are $30°$ and $45°$ respectively, find the height of the tree. (Use $\sqrt{3}=1.732$)

From the top of a $60 \mathrm{m}$ high building, the angles of depression of the top and the bottom of a tower are $45°$ and $60°$ respectively. Find the height of the tower. (Take $\sqrt{3}=1.732$)

In the given figure, from the top of a building $\mathrm{AB}$, $60 \mathrm{m}$ high, the angles of depression of the top and the bottom of a vertical lamp post $\mathrm{CD}$ are observed to be $30°$ and $60°$ respectively. Find the horizontal distance between $\mathrm{AB}$ and $\mathrm{CD}$.

In the given figure, from the top of a building $\mathrm{AB}$, $60 \mathrm{m}$ high, the angles of depression of the top and the bottom of a vertical lamp post $\mathrm{CD}$ are observed to be $30°$ and $60°$ respectively. Find the height of the lamp post $\mathrm{CD}$ (in $\mathrm{m}$).

In the given figure, from the top of a building $\mathrm{AB}$, $60 \mathrm{m}$ high, the angles of depression of the top and the bottom of a vertical lamp post $\mathrm{CD}$ are observed to be $30°$ and $60°$ respectively. Find the difference between the heights of the building and the lamp post (in $\mathrm{m}$).

A man standing on the deck of the ship which is $20 \mathrm{m}$ above the sea-level observes the angle of elevation of a bird as $30°$ and the angle of depression of its reflection in the sea as $60°$. Find the height of the bird from the sea level (in $\mathrm{m}$). Write the numerical value as the final answer.

A bird is sitting on the top of a $80 \mathrm{m}$ high tree. From a point on the ground, the angle of elevation of the bird is $45°$. The bird flew away horizontally in such a way that it remained at a constant height from the ground. After $2$ seconds, the angle of elevation of the bird from the same point is $30°$. Find the speed of flying of the bird. (Take $\sqrt{3}=1.732$)

From the top of a tower, the angles of depression of two objects on the same side of the tower are found to be $\alpha$ and $\beta \left(\alpha >\beta \right)$. If the distance between the objects is $p$, show that the height $h$ of the tower is given by $h=\frac{p\tan \alpha \tan \beta }{\tan \alpha -\tan \beta }$.