R. D. Sharma Solutions for Chapter: Height and Distance, Exercise 1: EXERCISE 12.1

A tower subtends an angle $\alpha$ at a point $A$ in the plane of its base and the angle of depression of the foot of the tower at a point $b$ metres just above $A$ is $\beta$. Prove that the height of the tower is $b \tan \alpha \cot \beta$.
 

An observer, $1.5 \mathrm{m}$ tall, is $28.5 \mathrm{m}$ away from a tower $30 \mathrm{m}$ high. If the angle of elevation of the top of the tower from his eye is $p°$, then find the value of $p$.

A carpenter makes stools for electricians with a square top of side $0.5 \mathrm{m}$ and at a height of $1.5 \mathrm{m}$ above the ground. Also, each leg is inclined at an angle of ${60}^{\mathrm{o}}$ to the ground. If the length of the longer step (where steps are at equal distance.) is $q \mathrm{m}$, then find the value of $q$, correct up to three decimal places. [$\mathrm{Take} \sqrt{3}=1.732$]

A boy is standing on the ground and flying a kite with $100 \mathrm{m}$ of string at an elevation of $30°$. Another boy is standing on the roof of a $10 \mathrm{m}$ high building and is flying his kite at an elevation of $45°$. Both the boys are on opposite sides of both the kites. Find the length of the string (in metres), rounded off to one decimal place, that the second boy must have so that the two kites meet. [Take $\sqrt{2}=1.414$]

From the top of a lighthouse, the angles of depression of two ships on the opposite sides of it are observed to be $\alpha$ and $\beta$. If the height of the lighthouse be $h$ metres and the line joining the ships passes through the foot of the lighthouse, show that the distance between the ship is $\frac{h(\tan \alpha +\tan \beta )}{\tan \alpha \tan \beta }$ meters.

From the top of a tower $\mathrm{h}$ metre high, the angles of depression of two objects, which are in the line with the foot of the tower are $\alpha$ and $\beta (\beta >\alpha )$. Find the value of $\frac{\mathrm{Distance} \mathrm{of} \mathrm{two} \mathrm{object}}{(\cot \alpha -\cot \beta )}$. 

A window of a house is $h$ metre above the ground. From the window, the angles of elevation and depression of the top and bottom of another house situated on the opposite side of the lane are found to be $\alpha$ and $\beta$ respectively. Prove that the height of the house is $h(1+\tan \alpha \tan \beta )$ metres.

The lower window of a house is at a height of $2 \mathrm{m}$ above the ground and its upper window is $4 \mathrm{m}$ vertically above the lower window. At certain instant the angles of elevation of a balloon from these windows are observed to be $60°$ and $30°$ respectively. If the height of the balloon above ground is $k \mathrm{m}$, then find the value of $k$.