Heights and Distances Questions and Answers

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Q21
From a top of a tower 100 m high, the angle of depression of the top of another tower is 30° and of the bottom is 45°. Find the height of the smaller tower.
  • A 42.26 m
  • B 56.7 m
  • C 50 m
  • D 57.74 m
Answer: Option D
Explanation: Let smaller tower height = h. Using tan relations: distance = 100 (from 45°), then 100 - h = 100 × tan 30° ⇒ h ≈ 57.74 m
Q22
A balloon is rising vertically at 5 m/s. An observer sees it at an elevation of 30° from a point 50√3 m away. Find the time taken for the angle to increase to 60°.
  • A 10 s
  • B 5 s
  • C 15 s
  • D 7 s
Answer: Option A
Explanation: Initial height = 50√3 × tan 30° = 50 m. Final height = 50√3 × tan 60° = 150 m. Height difference = 100 m. Time = 100/5 = 20 s
Q23
The angle of elevation to the top of a tower increases from 30° to 45° when moving 20 m closer. Find tower height.
  • A 20√3 m
  • B 40 m
  • C 17.32 m
  • D 30 m
Answer: Option C
Explanation: Let initial distance be x. Then tan 30° = h/x and tan 45° = h/(x-20). Solving gives h ≈ 17.32 m
Q24
From a point on the ground, the top of a tree is observed at 45° elevation. If the tree is 10 m tall, how far is the observer from it?
  • A 5 m
  • B 10 m
  • C 15 m
  • D 20 m
Answer: Option B
Explanation: tan 45° = height / base ⇒ 1 = 10/base ⇒ base = 10 m
Q25
Two observers standing at points A and B, 100 m apart, observe the top of a tower at angles 30° and 45°, respectively. Find the height of the tower.
  • A 57.7 m
  • B 86.6 m
  • C 50 m
  • D 75 m
Answer: Option A
Explanation: Let height = h. From A: tan 30° = h/x, from B: tan 45° = h/(100-x). Solving gives h ≈ 57.7 m
Q26
From the top of a 60 m high tower, the angle of depression of the top of a lamp post is 30° and that of its foot is 45°. Find the height of the lamp post.
  • A A) 34.64 m
  • B B) 38.92 m
  • C C) 42.36 m
  • D D) 30 m
Answer: Option A
Explanation: Use tan for both depressions, find base, then difference in heights.
Q27
The angle of elevation of a balloon from the ground is 45°. After ascending 120 m vertically, the elevation becomes 60°. Find the initial height of the balloon.
  • A A) 80 m
  • B B) 120 m
  • C C) 140 m
  • D D) 100 m
Answer: Option D
Explanation: Let height = h, base same for both cases.
Q28
Two towers stand 150 m apart. The angle of elevation of the top of the first as seen from the foot of the second is 30°, and that of the top of the second from the foot of the first is 60°. Find the ratio of their heights.
  • A A) 1:3
  • B B) 1:√3
  • C C) √3:1
  • D D) 2:3
Answer: Option C
Explanation: Use tan relations for each and find ratio.
Q29
The top of a tower subtends angles of elevation of 30° and 45° from two points in the same line at distances 100 m apart. Find the height of the tower.
  • A A) 100 m
  • B B) 73.2 m
  • C C) 50 m
  • D D) 60 m
Answer: Option B
Explanation: Let nearer point be x, use tan for both angles.
Q30
The angle of elevation of the top of a hill from a point on the ground is 45°. After walking 200 m toward the hill, the angle becomes 60°. Find the height of the hill.
  • A A) 346.4 m
  • B B) 273.2 m
  • C C) 200 m
  • D D) 400 m
Answer: Option B
Explanation: Use difference of distances and tan relations.

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