Some Applications of Trigonometry Exercise 12.1(part-3) | Extra Documents, Videos & Tests for Class 10 PDF Download

Q46.From the top of a 50 m high tower, the angles of depression of the top and bottom of a pole are observed to be 45^∘ and 60^∘ respectively. Find the height of the pole.

Soln:

AB = height of the tower = 50m.

CD = height of the pole

Angle of depression of top of building α=45^∘

Angle of depression of bottom of building β=60^∘

The above data is represented in the form of figure as shown

In right angle triangle if one of the included angle is Θ

AX = CX 

CD + AB – AX = 50 - 50/√3

Height of the building (pole) = 

Distance between the pole and tower = 50/√3 m

Q47.The horizontal distance between two trees of different heights is 60 m. the angles of depression of the top of the first tree when seen from the top of the second tree is 45^∘. If the height of the second tree is 80 m, find the height of the first tree.

Soln:

Distance between the trees = 60 m [BD]

Height of second tree = 80 m [CD]

Let height of the first tree = ‘h’ m [AB]

Angle of depression from second tree top to first tree top α=45^∘

The above data is represented in form of figure as shown

In right angle triangle if one of the included angle is Θ

CX = BD =60m

XB = CD = AB – AX

AX = 60 m

XB = CD = AX – AX

= 80 – 60

= 20 m

Height of the second tree = 80 m

Height of the first tree = 20 m

Q48.A flag staff stands on the top of a 5 m high tower. From a point on the ground, angle of elevation of the top of the flag staff is 60^∘ and from the same point, the angle of elevation of the top of the tower is 45^∘. Find the height of the flag staff.

Soln:

Height of tower = AB = 5

Height of flag staff BC = ‘h’ m

Angle of elevation of top of flagstaff α=60^∘

Angle of elevation of bottom of flagstaff β=45^∘

The above data is represented in form of figure as shown

In ΔABC

DA = 5 cm

In ΔADC

 3.65 mts

height of the flagstaff = 3.65 mts

 Q50.As observed from the top of a 150m tall light house, the angles of depression of two ships approaching it are 30^∘ and 45^∘. If one ship is directly behind the other, find the distance between the two ships.

Soln:

Height of the light house AB = 150 meters

Let S₁andS₂ be two ships approaching each other

Angle of depression of S₁,α=30^∘

Angle of depression of S₂,β=45^∘

Distance between ships = S₁S₂

The above data is represented in the form of figure as shown

In ΔABS₂

BS₂=150m

In ΔABS₁

BS₁=150√3

S₁S₂=BS₁–BS₂=150(√3–1 meters

Distance between ships = 150(√3−1 meters

Q51.The angle of elevation of the top of a rock form the top and foot of a 100 m high tower are respectively 30^∘ and 45^∘. Find the height of the rock

Soln:

Height of the tower AB = 100 m

Height of rock CD = ‘h’ m

Angle of elevation of the top of rock from top of the tower α=30^circ

Angle of elevation of the top of rock from bottom of tower β=60^circ

The above data is represented in the form of figure as shown

Draw AX⊥CD

XD = AB = 100m

XA = DB

In ΔCXA

DB = CX×sqrt3 —- (1)

In ΔCBD

DB = 100 + CX —- (2)

From (1) and (2)

100+CX=CX√3

100=CX(√3−1)

Height of the hill = 100+50(√3+1)

= 150(3+√3) meters

Q52.A straight highway leads to the foot of a tower of height 50m. From the top of the tower, the angles of depression of two cars standing on the highway are 30^∘ and 60^∘ respectively. What is the distance between the two cars and how far is each car from the tower?

Soln:

Height of the tower AB = 50m

C₁ and C₂ be two cars

Angles of depression of C₁ from top of the tower α=30^circ

Angles of depression of C₂ from top of the tower β=60^circ

Distance between cars C₁ and C₂

The above data is represented in form of figure as shown

 In ΔABC₂

In ΔABC₁

C₁ and C₂=BC₁–BC₂

Distance between cars 

Distance of car1 from tower = 50√3 meters

Distance of car2 from tower = 50/√3  meters

Q53.From the top of a building AB = 60m, the angles of depression of the top and bottom of a vertical lamp post CD are observed to be 30^∘ and 45^∘ respectively. Find

(i) The horizontal distance between AB and CD

(ii) The height of the lamp post

(iii) The difference between the heights of the building and the lamp post.

Soln:

Height of building AB = 60m

Height of lamp post CD = ‘h’ m

Angle of depression of top of lamp post from top of building α=30^circ

Angle of depression of bottom of lamp post from top of building β=60^circ

The above data is represented in form of figure

Draw DX ⊥ AB, CX = AC, CD = AX

In ΔBDX

AC = (60–h) √3m —- (1)

In ΔBCA

From (1) and (2)

(60–h)√3=20√ 3

60 – h = 20

h = 40m

Height of the lamp post = 40m

Distance between lamp post building AC = 20√3m

Difference between heights of building and lamp post BX = 60 – h => 60 – 40 = 20m.

Q54.Two boats approach a light house in mid sea from opposite directions. The angles of elevation of the top of the light house from two boats are 30^∘ and 45^∘ respectively. If the distance between the two boats is 100m, find the height of the light house.

Soln:

let B₁ be boat 1 and B₂ be boat 2

Height of light house = ‘h’ m = AB

Distance between B₁B₂=100m

Angle of elevation of A from B₁α=30^circ

Angle of elevation of B from B₂β=45^circ

The above data is represented in the form of figure as shown

Here In ΔABB₁

B₁B=AB√3 =h√3 —- (1)

In ΔABB₂

h = BB_{2} —- (2)

Adding (1) and (2)

Height of the light house = 50 ( √3-1 )

Q55.The angle of elevation of the top of a hill at the foot of a tower is 60⁰ and the angle of elevation of the top of the tower from the foot of the hill is 30⁰ . If tower is 50 m high, what is the height of the hill?

Soln:

Height of tower AB = 50m

Height of hill CD = ‘h’ m.

Angle of elevation of top of the hill from foot of the tower = α=60^∘

Angle of elevation of top of the tower from foot of the hill = β=30^∘

The above data is represented in the form of figure as shown

From figure

In ΔABC

BC = 50√3

In ΔBCD

CD = 150m

Height of hill = 150m.