Chapter 22 - Heights and Distances Exercise Ex. 22(B)

Question 1

In the figure, given below, it is given that AB is perpendicular to BD and is of length X metres. DC = 30 m, ADB = 30^o and ACB = 45^o. Without using tables, find X.

Solution 1

Question 2

Find the height of a tree when it is found that on walking away from it 20 m, in a horizontal line through its base, the elevation of its top changes from 60^o to 30^o.

Solution 2

Let AB be the tree of height h m.

Let the two points be C and D such that CD = 20 m, 

ADB = 30^o and ACB = 60^o

Hence, height of the tree is 17.32 m.

Question 3

Find the height of a building, when it is found that on walking towards it 40 m in a horizontal line through its base the angular elevation of its top changes from 30^o to 45^o.

Solution 3

Let AB be the building of height h m.

Let the two points be C and D such that CD = 40 m,

ADB = 30^o and ACB = 45^o

Hence, height of the building is 54.64 m.

Question 4

From the top of a light house 100 m high, the angles of depression of two ships are observed as 48^o and 36^o respectively. Find the distance between the two ships(in the nearest metre) if:

(i) the ships are on the same side of the light house.

(ii) the ships are on the opposite sides of the light house.

Solution 4

Let AB be the lighthouse.

Let the two ships be C and D such that 

ADB = 36^o and ACB = 48^o

(i) If the ships are on the same side of the light house,

then distance between the two ships = BD - BC = 48 m

(ii) If the ships are on the opposite sides of the light house,

then distance between the two ships = BD + BC = 228 m

Question 5

Two pillars of equal heights stand on either side of a roadway, which is 150 m wide. At a point in the roadway between the pillars the elevations of the tops of the pillars are 60^o and 30^o ; find the height of the pillars and the position of the point.

Solution 5

Let AB and CD be the two towers of height h m.

Let P be a point in the roadway BD such that BD = 150 m,

 APB = 60^o and CPD = 30^o

Hence, height of the pillars is 64.95 m.

The point is  from the first pillar.

That is the position of the point is  from the first pillar.

The position of the point is 37.5 m from the first pillar.

Question 6

From the figure, given below, calculate the length of CD.

Solution 6

Question 7

The angle of elevation of the top of a tower is observed to be 60^o. At a point, 30 m vertically above the first point of observation, the elevation is found to be 45^o. Find:

(i) the height of the tower,

(ii) its horizontal distance from the points of observation.

Solution 7

Let AB be the tower of height h m.

Let the two points be C and D such that CD = 30 m,

 ADE = 45^o and ACB = 60^o

Hence, height of the tower is 70.98 m

(ii)

The horizontal distance from the points of observation is BC = 40.98 m

Question 8

From the top of a cliff, 60 metres high, the angles of depression of the top and bottom of a tower are observed to be 30^o and 60^o. Find the height of the tower.

Solution 8

Let AB be the cliff and CD be the tower.

Here AB = 60 m, ADE = 30^o and ACB = 60^o

Hence, height of the tower is 40 m.

Question 9

A man on a cliff observes a boat, at an angle of depression 30^o, which is sailing towards the shore to the point immediately beneath him. Three minutes later, the angle of depression of the boat is found to be 60^o. Assuming that the boat sails at a uniform speed, determine:

(i) how much more time it will take to reach the shore.

(ii) the speed of the boat in metre per second, if the height of the cliff is 500 m.

Solution 9

Let AB be the cliff and C and D be the two positions of the boat such that 

ADE = 30^o 

and ACB = 60^o 

Let speed of the boat be x metre per minute and let the boat reach the shore after t 

minutes more.

Therefore, CD = 3x m ; BC = tx m

Hence, the boat takes an extra 1.5 minutes to reach the shore.

And, if the height of cliff is 500 m, the speed of the boat is 3.21 m/sec

Question 10

A man in a boat rowing away from a lighthouse 150 m high, takes 2 minutes to change the angle of elevation of the top of the lighthouse from 60^o to 45^o. Find the speed of the boat.

Solution 10

Let AB be the lighthouse and C and D be the

two positions of the boat such that 

AB = 150 m, ADB = 45^o and ACB = 60^o

Let speed of the boat be x metre per minute.

Therefore, CD = 2x m ;

Hence, the speed of the boat is 0.53 m/sec

Question 11

A person standing on the bank of a river observes that the angle of elevation of the top of a tree standing on the opposite bank is 60^o. When he moves 40 m away from the bank, he finds the angle of elevation to be 30^o. Find:

(i) the height of the tree, correct to 2 decimal places,

(ii) the width of the river.

Solution 11

Let AB be the tree of height 'h' m and BC be the width of the river. 

Let D be the  on the opposite bank of tree such that CD = 40 m.

Here  ADB = 30^o and ACB = 60^o

Let speed of the boat be x metre per minute.

Hence, height of the tree is 34.64 m and width of the river is 20 m.

Question 12

The horizontal distance between two towers is 75 m and the angular depression of the top of the first tower as seen from the top of the second, which is 160 m high, is 45^o. Find the height of the first tower.

Solution 12

Let AB and CD be the two towers

The height of the first tower is AB = 160 m

The horizontal distance between the two towers is

BD = 75 m

And the angle of depression of the first tower as seen from the top of the second

 tower is ACE = 45^o.

Hence, height of the other tower is 85 m

Question 13

The length of the shadow of a tower standing on level plane is found to be 2y metres longer when the sun's altitude is 30^o than when it was 45^o. Prove that the height of the tower is metres.

Solution 13

Let AB be the tower and C and D are two points such that 

CD = 2y m, ADB = 45^o 

and ACB = 30^o

Hence, height of the tower is  m.

Question 14

An aeroplane flying horizontally 1 km above the ground and going away from the observer is observed at an elevation of 60^o. After 10 seconds, its elevation is observed to be 30^o; find the uniform speed of the aeroplane in km per hour.

Solution 14

Let A be the aeroplane and B be the observer on the ground. The vertical height will be 

AC = 1 km = 1000 m. After 10 seconds, let the aeroplane be at point D.

Let the speed of the aeroplane be x m/sec

 CE = 10x

Hence, speed of the aeroplane is 415.69 km/hr

Question 15

From the top of a hill, the angles of depression of two consecutive kilometer stones, due east, are found to be 30^o and 45^o respectively. Find the distances of the two stones from the foot of the hill.

Solution 15

Let AB be the hill of height 'h' km and C and D be the

 two consecutive stones such that CD = 1 km,

 ACB = 30^o and ADB = 45^o.

Hence, the two stones are at a distance of 

1.366 km and 2.366 km from the foot of the hill.