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A straight highway leads to the foot of a 50 m tall tower. From the top of the tower, the angles of depression of two cars on the highway are $30 °$ and $60 °$ . What is the distance between the two cars and how far is each car from the tower?

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57m, 28m, 86m

57.76m, 28.88m, 86.6m

57.76m, 28.86m, 86.9m

57.74m, 28.86m, 86.6m

answer is D.

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Detailed Solution

Given that,
The angle of depression of the first car will be

θ = 60 °

The angle of depression of the second car will be

ϕ = 30 °

The height of the tower will be

h = 50

Δ P Q R

,
RQ (the distance from the first car to the tower) =

x

PQ=the tower
Substituting the values in

tan θ = h e i g h t b a s e

⇒ tan θ = h x ⇒ x = h tan θ

Substituting

θ

60 °

and

h

50

⇒ x = 50 tan 60 ° ⇒ x = 503 [∵ tan 60 ° = 3] ⇒ x = 28.86 m

Thus, the distance from the first car to the tower will be

28.86

m.
In

Δ P Q S

,
SR =y = the distance from the second car to the first car
SQ = the distance from the second car to the tower

x + y

= the distance SQ.

tan ϕ = P Q S Q ⇒ tan ϕ = P Q R Q + S R ⇒ tan ϕ = h x + y ⇒ x + y = h tan ϕ

Substituting

ϕ

30 °

and

h

50

⇒ x + y = 50 tan 30 ° ⇒ x + y = 5013 [∵ tan 30 ° = 13] ⇒ x + y = 503 ⇒ x + y = 86.6 m

Thus, the distance from the second car to the tower will be

86.6

m.
Substituting

x

28.86

m in

x + y = 86.6

⇒ 28.86 m + y = 86.6 m ⇒ y = 86.6 m- 28.86 m ⇒ y =57 .74 m

Thus, the distance between the two cars will be 57.74m, the distance from the first car to the tower will be

28.86

m and the distance from the second car to the tower will be

86.6

m.
Therefore the correct option is 4.

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