The angle of elevation of the top Q of a vertical tower P Q from a point X on the ground is 60 ° . At a point Y , 40 m vertically above X , the angle of elevation is 45 ° . Find the height of tower P Q .

We have X Y = 40 m , ∠ P X Q = 60 ° and ∠ M Y Q = 45 ° Let P Q = h Also, M P = X Y = 40 m , M Q = P Q - M P = h - 40 In Δ M Y Q , tan 45 ° = M Q M Y 1 = h - 40 M Y M Y = H - 40 P X = M Y = h . . . . . . . . . . . . . . . ( 1 ) Consider, In Δ P X Q , tan 60 ° = P Q P X 3 = h h - 40 [From i ] h 3 - 40 3 = h h 3 - h = 40 3 h 3 - 1 = 40 3 h = 40 3 3 - 1 h = 40 3 3 - 1 × 3 + 1 3 + 1 h = 40 3 3 + 1 3 - 1 h = 40 3 3 + 1 2 h = 20 3 3 + 1 h = 60 + 20 3 h = 60 + 20 × 1 . 73 h = 60 + 34 . 6 ∴ h = 64 . 6 m So, the height of the tower P Q is 94 . 6 m .

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 ° to 60 ° . Determine the distance travelled by the ship during the period of observation. [Take 3 = 1 . 732 ]

From the figure, height of the lighthouse AB = 100 m Let the ship is at point 'D' and it sails towards point 'C' let BC = x metres and CD = y metres Now in right-angled △ ABC , tan 60 ° = 100 x ⇒ 3 = 100 x Therefore, x = 100 3 metres ------(1) Now consider right-angled △ ABD , tan 30 ° = 100 x + y ⇒ 1 3 = 100 x + y ⇒ x + y = 100 3 Substitute the value of 'x' from (1) in the above equation, ⇒ 100 3 + y = 100 3 ⇒ y = 100 ( 3 - 1 3 ) Therefore, distance travelled by ship during the period of observation is, ⇒ y = 200 3 = 115 . 46 metres

MEDIUM

10th CBSE

IMPORTANT

Earn 100

A man on the deck of a ship, $16 m$ above water level, observe that angles of elevation and depression respectively of the top and bottom of a cliff are $60 °$ and $30 °$ . Calculate the distance of the cliff from the ship and height of the cliff. [Use $\sqrt{3} = 1.732$ ]

Important Questions on Heights and Distances

MEDIUM

10th CBSE

IMPORTANT

Secondary School Mathematics for Class 10>Chapter 14 - Heights and Distances>Exercise 14>Q 24

The angle of elevation of the top

Q

of a vertical tower

P Q

from a point

X

on the ground is

60 °

. At a point

Y

h = 48 + 16 = 64 m

vertically above

X

, the angle of elevation is

45 °

. Find the height of tower

P Q

MEDIUM

10th CBSE

IMPORTANT

Secondary School Mathematics for Class 10>Chapter 14 - Heights and Distances>Exercise 14>Q 25

The angle of elevation of an aeroplane from a point on the ground is

45 °

. After flying for

15

seconds, the elevation changes to

30 °

. If the aeroplane is flying at a height of

2500

meters, find the speed of the aeroplane.

MEDIUM

10th CBSE

IMPORTANT

Secondary School Mathematics for Class 10>Chapter 14 - Heights and Distances>Exercise 14>Q 26

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 °

. On advancing

150 m

towards foot of the tower, the angle of elevation becomes

60 °

. Show that the height of the tower is

129.9

metres.

MEDIUM

10th CBSE

IMPORTANT

Secondary School Mathematics for Class 10>Chapter 14 - Heights and Distances>Exercise 14>Q 27

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 °

60 °

. Determine the distance travelled by the ship during the period of observation. [Take

\sqrt{3} = 1.732

]

MEDIUM

10th CBSE

IMPORTANT

Secondary School Mathematics for Class 10>Chapter 14 - Heights and Distances>Exercise 14>Q 28

From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are

30 °

and

45 °

respectively. If the bridge is at a height of

2.5 m

from the banks, find the width of the river. [Use

\sqrt{3} = 1.732

]

MEDIUM

10th CBSE

IMPORTANT

Secondary School Mathematics for Class 10>Chapter 14 - Heights and Distances>Exercise 14>Q 29

The angles of elevation of the top of a tower from two points at a distance of undefined and undefined from the base of the tower and in the same straight line with it are complementary. Prove that the height of the tower is undefined.

MEDIUM

10th CBSE

IMPORTANT

Secondary School Mathematics for Class 10>Chapter 14 - Heights and Distances>Exercise 14>Q 30

A ladder of length

6 m

makes an angle of

45 °

with the floor while leaning against one wall of a room. If the fort of the ladder is kept fixed on the floor and it is made to lean against the opposite wall of the room, it makes an angle of

60 °

with the floor. Find the distance between two walls of the room.

HARD

10th CBSE

IMPORTANT

Secondary School Mathematics for Class 10>Chapter 14 - Heights and Distances>Exercise 14>Q 31

From the top of a vertical tower, the angles of depression of two cars in the same straight line with the base of the tower, at an instant are found to be $45 °$ and $60 °$ . If the cars are $100 m$ apart and are on the same side of the tower, find the height of the tower.

If height of the tower is $k m$ , find $k$ ?

A man on the deck of a ship, 16 m above water level, observe that angles of elevation and depression respectively of the top and bottom of a cliff are 60° and 30°. Calculate the distance of the cliff from the ship and height of the cliff. [Use 3=1.732]

Important Questions on Heights and Distances

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A man on the deck of a ship, $16 m$ above water level, observe that angles of elevation and depression respectively of the top and bottom of a cliff are $60 °$ and $30 °$ . Calculate the distance of the cliff from the ship and height of the cliff. [Use $\sqrt{3} = 1.732$ ]