Heights and Distances Questions and Answers

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Q31
From the top of a lighthouse 100 m high, the angle of depression of two ships is 30° and 45°. Find the distance between the ships.
  • A A) 57.7 m
  • B B) 73.2 m
  • C C) 100(√3 - 1) m
  • D D) 100(√3 + 1) m
Answer: Option C
Explanation: Find horizontal distances for both, subtract.
Q32
A man standing on a tower observes the top of a flagpole at an angle of depression of 30°. The flagpole is 10 m tall and is 40 m away from the tower's base. Find the height of the tower.
  • A A) 27.31 m
  • B B) 33.09 m
  • C C) 35.31 m
  • D D) 40 m
Answer: Option B
Explanation: Add both triangles' heights considering vertical alignment.
Q33
The angle of elevation of an aircraft from a point on the ground is 60°. After 10 seconds, it changes to 30°. If the plane is flying horizontally at a height of 2000 m, find its speed.
  • A A) 200√3 m/s
  • B B) 115.47 m/s
  • C C) 230.94 m/s
  • D D) 173.2 m/s
Answer: Option D
Explanation: Base = height/tan(angle), then find difference/time.
Q34
A flagstaff stands on top of a tower. The angles of elevation of the top and bottom of the flagstaff from a point on the ground are 45° and 30° respectively. If the point is 20 m away from the tower's base, find the height of the flagstaff.
  • A A) 8.64 m
  • B B) 11.55 m
  • C C) 12.36 m
  • D D) 10 m
Answer: Option B
Explanation: Subtract heights using tan values.
Q35
From a point 40 m from the base of a tower, its top is observed at 45° elevation. If the observer rises 10 m upward, the angle becomes 60°. Find the height of the tower.
  • A A) 35 m
  • B B) 46.18 m
  • C C) 40 m
  • D D) 50 m
Answer: Option B
Explanation: Consider height difference from new observation point.
Q36
From the top of a building 50 m high, the angle of depression to a car is 45°. After 10 seconds, the angle becomes 30°. Find the car's speed assuming it moves in a straight line.
  • A A) 14.4 m/s
  • B B) 12.1 m/s
  • C C) 10.5 m/s
  • D D) 8.3 m/s
Answer: Option A
Explanation: Use horizontal positions from both angles, subtract, divide by time.
Q37
The angles of elevation of the top of a tower from two points 100 m apart on level ground are 30° and 60°. Find the height of the tower.
  • A A) 86.6 m
  • B B) 100 m
  • C C) 50√3 m
  • D D) 75 m
Answer: Option A
Explanation: Use equations for both observations.
Q38
The top of a tower is observed from a point on a horizontal plane at an angle θ. If the observer moves 10 m closer, the elevation becomes complementary to the previous. Find height of the tower in terms of θ.
  • A A) 10 sin2θ
  • B B) 10 cos2θ
  • C C) 10 tanθ
  • D D) 10/(tanθ + tan(90° - θ))
Answer: Option A
Explanation: Use tan θ and tan(90° - θ) relations.
Q39
The shadow of a building increases by 10 m when the sun's elevation changes from 60° to 45°. Find the height of the building.
  • A A) 8.66 m
  • B B) 12.32 m
  • C C) 17.32 m
  • D D) 10 m
Answer: Option C
Explanation: Use tan for both angles and subtract.
Q40
The angle of elevation of the top of a tower at a distance of x m is tan⁻¹(3/4). Find the height of the tower.
  • A A) 4x/3
  • B B) 3x/4
  • C C) x
  • D D) 5x/4
Answer: Option B
Explanation: tan θ = height/base.

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