Calculating Heights and Distances

Q.

The shadow of a tower standing on a level ground is found to be 40 m longer when the Sun's altitude is 30^o than when it is 60^o. Find the height of the tower.

1. 20

2. 40

3. 40√3

4. 20√3

Q.

The angle of elevation of an aeroplane from a point on the ground is ${45}^{\circ }$. After a flight of 15 seconds, the angle of elevation changes to ${30}^{\circ }$. If the aeroplane is flying at a height of 3000 m, then find the speed of the plane.

$(\text{Take}\sqrt{3}=1.732)$

1. 150 km/h

2. 146.4 km/h

3. 146.4 m/s

4. 150 m/s

Q.

Two poles of equal heights are standing opposite to each other on either side of the road which is 80 m wide. From a point P between them on the road, the angle of elevation of the top of one pole is ${60}^{\circ }$ and the angle of depression from the top of another pole at P is ${30}^{\circ }$. Find the height of each pole and distances of the point P from the poles.

Q.

From the top of a hill, the angles of depression of two consecutive kilometre stones due east are found to be ${45}^{\circ }$ and ${30}^{\circ }$ respectively. Find the height of the hill.

Q.

From a point on a bridge across a river the angles of depression of the banks on opposite sides of the river are ${30}^{\circ }$ and ${45}^{\circ }$ respectively. If the bridge is at a height of 6 m from the banks find the width of the river.

1. 6 m

2. 6√3 m

3. 3√3 m

4. 6+ 6√3 m

Q. Sec theta(1-sin theta)(sec theta+ tan theta)=1

Q.

From a point on the ground, the angles of elevation of the bottom and top of a transmission tower fixed at the top of $20m$ high building are ${45}^{\circ }and{60}^{\circ }$ respectively. Find the height of the transmission tower.

Q. If a 1.5 m tall girl stands at a distance of 3 m from a lamp-post and casts a shadow of length 4.5 m on the ground, then the height of the lamp-post is

(a) 1.5 m

(b) 2 m

(c) 2.5 m

(d) 2.8 m

Q.

What are the full forms of $\sin ,\cos ,\sec ,\mathrm{cosec},\tan$and $\cot$?

Q.

A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle of ${30}^{\circ }$ with the ground. The distance between the foot of the tree to the point where the top touches the ground is 8m. Find the height of the tree.

Q.

The angle of elevation of the top of a building from the foot of a tower is ${30}^{\circ }$. The angle of elevation of the top of the tower from the foot of the building is ${60}^{\circ }$. If the tower is 60 m high, find the height of the building.

Q.

Question 13
As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are ${30}^{\circ }$ and ${45}^{\circ }$. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.

Q.

The angle of elevation of the top of a tower from a point on the same level as the foot of the tower is ${30}^{\circ }$. On advancing 150 m towards the foot of the tower, the angle of elevation becomes ${60}^{\circ }$. Show that the height of the tower is 129.9 metres. [Take $\sqrt{3}=\mathrm{1.732.}$]

Q.

From the top of a building 60 m high, the angles of depression of the top and bottom of a tower are observed to be ${30}^{\circ }$ and ${60}^{\circ }$. Find the height of the tower.

Q.

Question 6
A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eyes to the top of the building increases from ${30}^{\circ }to{60}^{\circ }$ as he walks towards the building. Find the distance he walked towards the building.

Q.

The angle of depression from the top of a tower of a point A on the ground is ${30}^{\circ }$ . On moving a distance of 20 metres from the point A towards the foot of the tower to a point B, the angle of elevation of the top of the tower from the point B is ${60}^{\circ }$. Find the height of the tower and its distance from the point A.

Q.

A TV tower stands vertically on a bank of canal. From a point on the other bank directly opposite the tower, the angle of elevation of the top of the tower is ${60}^{\circ }$. From another point 20 m away from this point on the line joining this point to the foot of the tower, the angle of elevation of the top of the tower is ${30}^{\circ }$. Find the height of the tower and the width of the canal.

Q. A man observes a car from the top of a tower, which is moving towards the tower with a uniform speed. If the angle of depression of the car changes from ${30}^{\circ }$ to ${45}^{\circ }$ in 12 minutes, find the time taken by the car now to reach the tower.

Q.

An observer 1.5 m tall is 28.5 m away from a tower and the angle of elevation of the top of the tower from the eye of the observer is ${45}^{\circ }$. The height of the tower is

(a) 27 m

(b) 30 m

(c) 28.5 m

(d) none of these

Q.

A TV transmission tower antenna is at a height of $20m$. Suppose that the receiving antenna is at.

(i) Ground level

(ii) a height of $5m$

The increase in antenna range in case (ii) relative to case (i) is $n\%$. The value of $n$, to the nearest integer, is

Q. An aeroplane when flying at a height of 3000 metres from the ground passes vertically above another aeroplane at an instant when the angles of elevation of the two planes from the same point on the ground are ${60}^{\circ }$ and ${45}^{\circ }$ respectively. Find the vertical distance between the aeroplanes at that instant. [Take $\sqrt{3}=1.73$] [3 MARKS]

Q.

A tree $12m$ high, is broken by the wind in such a way that its top touches the ground and makes an angle ${60}^{\circ }$ with the ground. At what height from the bottom, the tree is broken by the wind?

Q.

If the angle of elevation of a cloud from a point h metres above a lake is a and the angle of depression of its reflection in the lake be b, prove that the distance of the cloud from the point of observation is $\frac{2hsec\alpha }{tan\beta -tan\alpha }$

Q.

The angle of elevation of a jet fighter from a point A on the ground is ${60}^{\circ }$. After a flight of 15 seconds, the angle of elevation changes to ${30}^{\circ }$. If the jet is flying at a speed of 720 km/hour, find the constant height at which the jet is flying. [Use $\sqrt{3}=1.732$]

1. 3598 m

2. 2598 m

3. 3098 m

4. 2798 m

Q. At a point on level ground, the angle of elevation of a vertical tower is found to be such that its tangent is 5/12.On walking 192m towards the tower, the tangent of the angle becomes 3/4.Find the height of the tower.

Q.

As observed from the top of a lighthouse, 100 m above sea level, the angle of depression of a ship, sailing directly towards it, changes from ${30}^{\circ }to{60}^{\circ }$. Determine the distance travelled by the ship during the period of observation. [Take $\sqrt{3}=\mathrm{1.732.}$]

Q.

From the top of a building 60 m high, the angles of depression of the top and bottom of a tower are observed to be ${30}^{\circ }$ and ${60}^{\circ }$. Find the height of the tower.

1. 80 m

2. 90 m

3. 70 m

4. 40 m

Q. The angle of elevation of a cloud from a point h metres above a lake is $\alpha$ and the angle of depression of its reflection in the lake is $\beta$. Prove that the height of the cloud is $\frac{h(tan\beta +tan\alpha )}{(tan\beta -tan\alpha )}metres$ [4 MARKS]

Q.

A statue 1.6 m tall stands on the top of pedestal. From a point on the ground, the angle of elevation of the top of the statue is ${60}^{\circ }$ and from the same point the angle of elevation of the top of the pedestal is ${45}^{\circ }$. Find the height of the pedestal.

Q.

A vertical tower stands on a horizontal plane and is surmounted by a vertical flag-staff. At a point on the plane 70 metres away from the tower, an observer notices that the angles of elevation of the top and the bottom of the flag-staff are respectively ${60}^{\circ }and{45}^{\circ }$. Find the height of the flag-staff and that of the tower.