Practice question on height and distance for IIT JEE , Airforce and NDA 16-30 (SET 2 )

Height and distance

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Question no. 16

The angle of depression of a ship from the top of a tower 30 m high is 60^o , then the distance of ship from the base of tower is

looks_one 30 m

looks_two 30√3 m

looks_3 10√3 m

looks_4 10 m

Option looks_3 10√3 m

Solution :

In given triangle
tan 60^o = 30/x
√3 = 30/x
x = 30/√3
x = 10√3

Question no. 17

A tree is broken by wind , its upper part touches the ground at a point 10 meters from the foot of the tree and makes an angles of 45^o with the ground , the total length of tree is

looks_one 15

looks_two 20

looks_3 10(1+√2)

looks_4 10(1+ √3/2)

Option looks_3 10(1+√2)

Solution :

In given triangle
total height = AB + BC
tan45^o = BC/AC
BC = 10
AB = 10√2
Total height = AB + BC = 10(1+√2)

Question no. 18

A Tower Subtends an angle of 30^o at a point distant d from the foot of the tower and on the same level as the foot of the tower. At a second point h vertically above the first, the depression of the foot of the tower is 60^o. The height of the tower is

looks_one h/3

looks_two h/3d

looks_3 3h

looks_4 3h/d

option looks_one h/3

Solution :


In given triangle
tan30^o = H/x .......[1]
tan60^o = h/x .....[2]
dividing [1] from [2]
tan30^o/ tan60^o = H/h
1/3 = H/h
h = 3H
H = h/3

Question no. 19

The angle of elevation of the sun, when the shadow of the pole √3 times the height of the pole

looks_one 60^o

looks_two 30^o

looks_3 45^o

looks_4 15^o

option looks_two 30^o

Solution :

In △ XYZ
tanθ = h/x
tanθ = h/√3h
tanθ = 1/√3
θ = 30^o

Question no. 20

A house subtends a right angle at the window of an opposite house and the angle of elevation of the window from the bottom of the first house is 60^o. If the distance between the two houses be 6 metres ,then the height of the first house is

looks_one 6√3

looks_two 8√3

looks_3 4√3

looks_4 none of these

Option looks_one 6√3

Solution :

tan 60^o = h/x
√3 = h/6
h = 6√3

Question no. 21

The angle of elevation of the top of a flag post from a point 5 m away from its base is 75^o What is the approximate height of flag post?

looks_one 15 m

looks_two 17 m

looks_3 19 m

looks_4 21 m

option looks_3 19 m

Solution :

In △ ABC
tan75^o = AB/BC

Question no. 22

Looking from the top of a 20 m high building ,the angle of elevation of the top of a tower is 60^o and the angle of depression of it bottom is 30^o. What is the height of the tower?

looks_one 50 m

looks_two 60 m

looks_3 70 m

looks_4 80 m

option looks_4 80 m

Solution :

In △ BDE
tan 30^o = BD/DE = 20/x
x = 20√3
In △ BAC
tan 60^o = AC/BC = h/x
h = √3
h = √3 (20√3) = 60 m
total height = AC + EC = 60 + 20 = 80 m

Question no. 23

Two straight roads intersect at an angle of 60^o. A bus on one road is 2 km away from the intersection and a car on the other road is 3 km away from the intersection. Then the direct distance between two vehicles is

looks_one 1 km

looks_two √2 km

looks_3 4 km

looks_4 √7 km

option looks_4 √7 km

Solution :

Question no. 24

AB is a vertical pole resting at the end A on the level ground. P is a point on the level ground such that AP=3AB . If C is the midpoint of AB and CB subtends an angle b at p the value of tan b is

looks_one 18/19

looks_two 3/19

looks_3 1/6

looks_4 none of these

option looks_3 1/6

Solution :

AP/AB = 3
tan b = AC/AP
tan b = 0.5AB/AP = 0.5/3 = 1/6

Question no. 25

A person standing on the bank of a river finds that the angle of elevation of the top of a tower on the opposite bunk is 45^o. Then which of the following statements is correct

looks_one Breath of the river is twice the height of the tower

looks_two Breadth of the river and the height of the tower are the same

looks_3 breadth tower of the river half of the height of the tower

looks_4 none of these

option looks_two Breadth of the river and the height of the tower are the same

Solution :

In △ XYZ
tan 45^o = YZ/XZ = h/B
1 = h/B
h = B
Breadth of the river and the height of the tower are the same

Question no. 26

A balloon is coming down at the 4 m/ min and its angle of elevation is 45^o from a point on the ground which has been reduced to 30^o after 10 minute. Balloon will be on the ground at distance of how many meters from the observer?

looks_one 20√3

looks_two 20(√3 + 3)

looks_3 10(√3 + 3)

looks_4 none of these

option looks_two 20(√3 + 3)

Solution :

In △ ABD
tan 45^o = BD/AD
1 = (40 + h)/x
In △ ACD
tan 30^o = CD/AD
1/√3 = h/x
x = √3h
x = 40 + h
x = 40 + x/√3
√3x = 40√3 + x
x = 40√3/(√3 -1) = 20√3(1 + √3)

Question no. 27

From the top of a lighthouse 120 m above the sea, the angle of depression of a boat is 15^o. What is the distance of the boat from the lighthouse?

looks_one 400 m

looks_two 421 m

looks_3 444 m

looks_4 460 m

option looks_3 444 m

Solution :

In △ ABC
tan 15⁰ = 120/x

Question no. 28

The angle of elevation of the top of a tower from a point 20 m away from its base is 45^o. What is the height of the tower?

looks_one 10 m

looks_two 20 m

looks_3 30 m

looks_4 40 m

option looks_two 20 m

Solution :

In △ ABC
tan 45^o = h/20
h = 20

Question no. 29

The shadow of a tower standing on a level plane is found to be 50 m longer when the sun's elevation is 30^o than when it is 60^o. The height of the tower is

looks_one 25 m

looks_two 25√3 m

looks_3 50 m

looks_4 none of these

option looks_two 25√3 m

Solution :

In △ ABC
tan 60^o = h/x
√3 = h/x
h = √3x
In △ ABD
tan 30^o = h/(x+50)
1/√3 = h/(x+50)
x+50 = √3h
x + 50 =√3 (√3x)
x+50 = 3x
2x = 50
x = 25
h = √3x = 25√3

Question no. 30

The top of a hill observed from the top and bottom of a building of height h is at the angles of elevation a and b. The height of hill is

looks_one hcot(b)/(tan b - tan a)

looks_two hcot(a)/(tan a - tan b)

looks_3 htan(a)/(tan a - tan b)

looks_4 none of these

option looks_4 none of these

Solution :

In △ABE
tan a = (H-h)/x ......[1]
In △ADC
tan b = H/x ........[2]
Dividing the equation [1] and [2]
tan a / tan b = (H-h)/H
H(tan a) =(H -h) tan b
H =( h tanb)/(tan b - tan a)

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