Heights & Distances

Q.

At the foot of a mountain the elevation of its summit is ${45}^{\circ }$, after ascending 1000 m towards the mountain up a stop of ${30}^{\circ }$ inclination, the elevation is found to be ${60}^{\circ }$. Find the height of the mountain :

1. 1.3 km

2. 1.366 km

3. 2.72 km

4. None of these

Q.

If the shadow of a tower is 30 m when the sun's altitude is ${30}^{\circ }$ what is the length of the shadow when the Sun's altitude is ${60}^{\circ }$?

1. 20 m

2. $10\sqrt{3}m$

3. 10 m

4. 12 m

Q. Two towers of the same height stand on either side of a read 60 m wide. At a point on the road between the towers, the elevations of the towers are ${60}^{\circ }$ and ${30}^{\circ }$. Find the approximate height of the towers. ___

Q. A ladder leans against a vertical wall. The top of the ladder is 8m above the ground. When the bottom of the ladder is moved 2m farther away from the wall, the top of the ladder rests against the foot of the wall. What is the length of the ladder ?

1. 20m
2. 17m
3. 15m
4. 10m

Q. An aeroplane when 3000 m high passes vertically above another aeroplane at an instant when their angles of elevation at the same observing point are ${60}^{\circ }$ and ${45}^{\circ }$ respectively. How many metres higher is the one than the other?

1. 1752 m
2. 1268 m
3. 1248 m
4. 1188 m

Q. From the top of a cliff 100 m high, the angles of depression of the top and the bottom of a tower are observed to be ${30}^{\circ }$ and ${60}^{\circ }$, respectively. The height of the tower is

1. 56.8 m
2. 23.7 m
3. None of these
4. 66.7 m

Q.

From the top of a cliff, 200 m high, the angle 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 (in fraction; in meter)___

Q.

Two towers of the same height stand on opposite sides of a road 100 m wide. At a point on the road between the towers, the elevations of the towers are ${30}^{\circ }$ and ${45}^{\circ }$ Find the height of the towers and the position of the point from one of the towers:

1. 36.6 m and 63.4 m

2. 63.6 m and 63.4 m

3. 66.3 m and 63.4 m

4. 36.6 m and 86.4 m

Q.

A man standing at a certain distance from a building, observes the angle of elevation of its top as ${60}^{\circ }$. He walks 30 meters away from the building. Now, the angle of elevation of the building's top is ${30}^{\circ }$. How high is the building?

1. 15 m

2. 30 m

3. 153 m

4. 15(3-1) m

5. 53 m

Q.

A man standing on the deck of a ship, which is 10m above the water level, observes the angle of elevation of the top of a hill as 60 degrees and the angle of depression of the base of the hill as 30 degrees. Find the distance of the hill from the ship and the height of the hill.

Q. The angles of elevation of an artificial satellite as measured from two earth stations are ${30}^{\circ }$ and ${60}^{\circ }$. If the distance between the earth stations is 4000 km, the height of the satellite is

1. 3152 km
2. 2000 km
3. 3465 km
4. 2800 km

Q.

The angle of elevation of a cloud from a height h above the level of water in a lake is $\alpha$ and the angle of depression of its image in the lake is $\beta$. Find the height of the cloud above the surface of the lake :

1. $\frac{hsin(\beta -\alpha )}{sin(\alpha +\beta )}$

2. $hsin\alpha$

3. $\frac{hsin(\alpha +\beta )}{sin(\beta -\alpha )}$

4. None of these

Q.

A man on the top of a rock rising on a seashore observes a boat coming towards it. If it takes 10 minutes for the angle of depression to change from ${30}^{\circ }$ to ${60}^{\circ }$, how soon the boat reaches the shore?

1. 5 min

2. 4 min

3. 10 min

4. 3 min

Q. The moon's distance from the earth is $360000$ km and its diameter subtends an angle of ${42}^{\prime }$ at the eye of the observer. The diameter of the moon is?

1. $3600$ km
2. $4400$ km
3. $8800$ km
4. $1000$ km

Q.

A man is walking along a straight road. He notices the top of a tower subtending an angle A = 60${}^{\circ }$ with the ground at the point where he is standing. If the height of the tower is h = 30 m, then what is the distance (in meters) of the man from the tower (Use $\surd$3 = 1.732)?

1. 17.32

2. 20

3. 25.32

4. 15.23

5. None of these

Q. A plan tree 90 ft high, is broken by the wind and its upper part meet the ground at an angle of ${30}^{\circ }$. Find the distance of the point where the top of the tree meets the ground, from its root:

1. 51.96 ft
2. 43.69 ft
3. 60 ft
4. 30 ft

Q.

The elevation of a tower at a station A due north of it is ${45}^{\circ }$ and at a station B due west of A is ${30}^{\circ }$. If AB = 40 m, find the height of the tower:

1. 28.28 m

2. 38.5 m

3. None of these

4. 26.26 m

Q. A boy is standing on the ground and flying a kite with 120 m of string at an elevation of ${30}^{\circ }$. Another boy is standing on the roof of a 20 m high building and is flying his kite at an elevation of ${60}^{\circ }$. The kites are flying towards each other and the boys are on opposite sides of both the kites. The length of string that the second boy must have so that the two kites meet is

1. $80\frac{\sqrt{3}}{3}$
2. $70\sqrt{3}m$
3. $48\frac{\sqrt{3}}{3}$
4. $15\sqrt{3}m$

Q. A helicopter is in a stationary position at a certain height over the lake. At a point 200 m above the surface of the lake, the angle of elevation of the helicopter is ${45}^{\circ }$. At the same time, the angle of depression of its reflection in the lake is ${75}^{\circ }$. Calculate the height of the helicopter from the surface of the lake:

1. 150 m
2. 200 m
3. $200\sqrt{3}m$
4. $\frac{200}{\sqrt{3}}m$

Q.

Anil looked up at the top of a lighthouse from his boat, and found the angle of elevation to be 30${}^{\circ }$. After sailing in a straight line 50 m towards the lighthouse, he found that the angle of elevation changed to 45${}^{\circ }$. Find the height of the lighthouse.

1. 25

2. 25(√3+1)

3. 25√3

4. 25(√3-1)

Q. A ladder leans against a vertical wall. The top of the ladder is 8m above the ground. When the bottom of the ladder is moved 2m farther away from the wall, the top of the ladder rests against the foot of the wall. What is the length of the ladder ?
(CAT 2001)

1. 10m
2. 15m
3. 20m
4. 17m

Q.

There is a small isle in the middle of a 200m wide stream and a tall palm tree stands on the isle. A and B are points directly opposite to each other on two banks and in line with the palm tree. The angles of elevation to the top of the palm tree from A and B are respectively 30${}^0$ and 45${}^0$.
What is the height of the palm tree?

1. 113.2 m

2. 73.2 m

3. None of these

4. 53.2 m

5. 93.2 m

Q.

Two towers of equal height are on either side of a river, which is 200 m wide. The angles of elevation of the top of the towers are ${60}^{\circ }$ and ${30}^{\circ }$ at a point on the river between the towers. Find the position of the point between the towers.

1. 50 m and 150 m

2. 60 m and 140 m

3. 90 m and 110 m

4. 70 m and 130 m

5. 34 m and 166 m

Q.

A hunter spots a bird flying in the sky at an elevation 60${}^{\circ }$ from his line of sight. Keeping in mind the speed of the bird, he fires his gun at an elevation 45${}^{\circ }$ with his line of sight. However, the bird changes its direction by the time the bullet crosses the bird, the bird has flown down to an elevation 30º from his line of sight. At the time the bullet and the bird are vertically in-line, what would be the distance between them (Assume initial height of the bird from the ground = 2000m)?

1. 
   

2. 
   

3. 
   

4. 
   

Q. The angle of elevation of the top of a chimney is ${45}^{\circ }$ from a point on the ground at a distance of 200 metres from the base of the chimney. The height of the chimney is

1. $100\sqrt{2}m$
2. $200m$
3. $50\sqrt{2}m$
4. $200\sqrt{2}m$

Q. A ladder is resting with one end in contact with the top of a wall of height 12 m and the other end on the ground is at a distance 5 m from the wall. The length of the ladder is _____.

1. 13 m
2. 17 m
3. 16 m
4. 18 m
5. None of these

Q.

Two poles of equal heights are standing opposite each other on either side of the road, which is $80\mathrm{m}$ wide.

From a point between them on the road, the angles of elevation of the top of the poles are $60°$ and $30°$, respectively.

Find the height of the poles and the distances of the point from the poles.

Q. In triangle ABC, $\angle C={90}^{\circ }.$ If inradius = r and circumradius = R, then find 2(r + R)?(a, b, c are the sides of the triangle opposite to angles A, B and C respectively)

1. a + b
2. c + b
3. a + c
4. a + b + c

Q.

Each corner of a square subtends an angle of 30${}^{\circ }$ at the top of a tower 'h' meters high standing in the centre of the square. If 'a' is the length of the each side of square then the relation between h and a is

1. 2h² = 3a²

2. 2a² = h²

3. 2h² = a²

4. 3a² =2h²

Q. A round balloon of radius 'r' subtends an angle $\alpha$ at the eye of the observer, while the angle of elevation of its centre is $\beta$. Find the height of the centre of balloon.

1. $rcosec\left(\frac{\alpha }{2}\right)sin\beta$
2. $rsin\alpha cosec\beta$
3. $\frac{r}{2}sin\beta$
4. $rsec\left(\frac{\alpha }{2}\right)$