Heights and Distances

An aeroplane when 900 m high passes vertically above another aeroplane at an instant when their angles of elevation at same observing point are 60° and 45° respectively. Approximately, how many meters higher is the one than the other?

The angle of elevation of an aeroplane from a point on the ground is $45°$, after $15 \sec$ of flight, the elevation changes to $30°$, if the aeroplane is flying at a height of $3000 \mathrm{m}$, find the speed of the plane.

The angle of elevation of the top of a tower from a point on the ground is ${45}^{°}$. Moving $21 \mathrm{m}$ directly towards the base of the tower, the angle of elevation changes to ${60}^{°}$.  What is the height of the tower to the nearest meter?

At a point $P$ on the ground, the angle of elevation of the top of a $10\mathrm{m}$ tall building and of a helicopter hovering some distance over the top of the building are $30º$ and $60º$ respectively. Then, the height of the helicopter above the ground is

What is the angle of elevation of the sun when the length of the shadow of a vertical pole is equal to its height?

At a point $A$, the angle of elevation of a tower is found to be such that its tangent is $\frac{ 5}{12}$ . On walking $192 \mathrm{meter}$ towards the tower, the tangent of the angle of elevation is found to be $\frac{3}{4}$ . Find the height of the tower.the

The angle of depression of two ships from the top of a lighthouse are $45°$ and $30°$ towards the east. If two ships are $200 \mathrm{m}$ apart. Find the height of the lighthouse.

A man moves $\sqrt{2}x \mathrm{km}$ East from his residence and then moves $x \mathrm{km}$ North. He then goes $x \mathrm{km}$North-East and finally he takes a turn of $90°$ towards right and moves a distance $x \mathrm{km}$and reaches his office. What is the shortest distance of the office from his residence?

The angle of elevation of a boat from a $15 \mathrm{m}$ high bridge is $30°$. If boat is moving with the speed of $6 \mathrm{km}/\mathrm{hr},$ then time taken by the boat to get under the bridge is

The angle of elevation of the top of a tower from certain point is ${30}^{\circ }$. If the observer moves $20 \mathrm{m}$ towards the tower, the angle of elevation of the top increases by ${15}^{\circ }$. Find the height of the tower?

Two ships are sailing in the sea on the two sides of a lighthouse. The angle of elevation of the top of the lighthouse is observed from the ships are $30°$ and $45°$ respectively. If the lighthouse is $100 \mathrm{m}$ high, Find the distance between the two ships.

A person stands at a height of $10 \mathrm{m}$ wants to get a fruit which is on a pole of height $\frac{50}{3} \mathrm{m}$ . If he stands at a distance of $\frac{20}{\sqrt{3} } \mathrm{m}$ from the foot of the pole horizontally, then the angle at which he should throw the stone, so that it hits the fruit is ______.

A telegraph post gets broken at a point against a storm and its top touches the ground at a distance $20 \mathrm{m}$ from the base of the post making an angle ${30}^{°}$ with the ground. What is the height of the post (in $\mathrm{m}$)?

The horizontal distance between two towers is $60 m$. The angular elevation of the top of the taller tower as seen from the top of the shorter one is $30°$. If the height of the taller tower is $150 m$, the height of the shorter one approximately equal to_____.

The angle of elevation of a cloud from a point 'h' meter above a lake is $\varnothing$. The angle of depression of its reflection in the lake is $45°$. The height of the cloud is equal to _____.

A man is watching form the top of the tower a boat speeding away from the tower. The boat makes the angle of depression of $45°$ with the man's eye when at a distance of$60 \mathrm{m}$ from the tower. After $5 \sec$ the angle of depression becomes $30°$. What is the approximate speed of the boat, assuming that it is running in still water ?

The angle of elevation of the top of a tower from a point on the ground, which is$30\mathrm{m}$ away from the foot of the tower is $30$°. Find the height of the tower.

A man is watching from the top of a tower a boat speeding away from the tower. The boat makes an angle of depression of ${45}^{°}$ with the man's eye when at a distance of $60 \mathrm{m}$ from the bottom of tower. After $5 \mathrm{seconds}$, the angle of depression becomes ${30}^{°}$. What is the approximate speed of the boat assuming that it is running in still water?

If the angles of elevation of a balloon from two consecutive stones in $1 \mathrm{km}$ distance along a road are $30°$ and $60°$ respectively, then find the height of the balloon above the ground.

The top of a $15$$m$ high tower makes an angle of elevation of ${60}^0$ with the bottom of an electric pole and an angle of elevation of ${30}^0$ with the top of the pole. Find the height of the electric pole?