Question

1. A man, standing at the foot of a hillock, observed the
top of the hillock at an angle of elevation of 45°. There
is a straight path towards the top. The path makes an
angle of 30° with the horizontal. After covering a
distance d along this path, the man sees the top of
the hillock at an angle of elevation of 60°. If the height
of the hillock is 40 m find the distance the man covers
along the path.
​

Answer 1

Complete Question :- A man, standing at the foot of a hillock, observed the top of the hillock at an angle of elevation of 45°. There is a straight path towards the top. The path makes an angle of 30° with the horizontal. After covering a distance d along this path, the man sees the top of the hillock at an angle of elevation of 60°. If the height of the hillock is 40 m . find the distance the man covers along the path. ?

Answer :-

In right ∆DEC we have,

→ sin 30° = DE/DC

→ (1/2) = DE/d

→ DE = (d/2) --------- Eqn.(1)

and,

→ cos 30° = EC/DC

→ (√3/2) = EC/DC

→ (√3/2) = EC/d

→ EC = (√3d/2) -------- Eqn.(2)

now, in rectangle FDEB, we have,

FD = BE

FB = DE

also,

→ AF = AB - FB

→ AF = (40 - DE)

→ AF = (40 - d/2)

then,in right angled ∆AFD,

→ tan 60° = AF/FD

→ √3 = (80 - d)/2FD

→ FD = (80 - d)/2√3

→ BE = (80 - d)/2√3 --------- Eqn.(3)

now, in ∆ABC,

→ tan 45° = AB/BC

→ 1 = 40/BC

→ BC = 40

→ BE + EC = 40

putting values of Eqn.(2) and Eqn.(3)

→ (80 - d)/2√3 + (√3d/2) = 40

→ (80 - d + 3d)/2√3 = 40

→ 80 + 2d = 80√3

→ 2d = 80√3 - 80

→ 2d = 80(√3 - 1)

→ d = 40(√3 - 1) (Ans.)

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