Heights and Distances Questions and Answers

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Q11
The angle of elevation of a tower from a point on the ground is 45°. On moving 10 m closer, the angle becomes 60°. Find the height of the tower.
  • A 10√3 m
  • B 15√3 m
  • C 20 m
  • D 20√3 m
Answer: Option A
Explanation: Let initial distance be x. Then tan 45° = h/x and tan 60° = h/(x-10). Solving gives h = 10√3 m
Q12
An observer 1.6 m tall sees the top of a tower at an elevation of 30°. The distance between the observer and the tower is 20√3 m. Find the height of the tower.
  • A 12 m
  • B 11.6 m
  • C 10 m
  • D 15 m
Answer: Option B
Explanation: Height = (tan 30° × base) + observer's height = (1/√3 × 20√3) + 1.6 = 20 + 1.6 = 21.6 m
Q13
A building 50 m high casts a shadow of 50√3 m. Find the angle of elevation of the sun.
  • A 60°
  • B 30°
  • C 45°
  • D 75°
Answer: Option B
Explanation: tan θ = height / shadow = 50/(50√3) = 1/√3 ⇒ θ = 30°
Q14
A hill is observed at an elevation of 45° from a point A. After ascending 100 m vertically to point B, the angle becomes 60°. Find the height of the hill.
  • A 273.2 m
  • B 300 m
  • C 273 m
  • D 200 m
Answer: Option C
Explanation: Let height = h. Using geometric relations: tan 45° = h/d and tan 60° = (h-100)/d. Solving gives h ≈ 273 m
Q15
The angle between an observer's line of sight to the top of a tower and the horizontal is 45°. If the observer is 40 m from the base, find the tower's height.
  • A 40 m
  • B 20√3 m
  • C 60 m
  • D 30 m
Answer: Option A
Explanation: tan 45° = height / distance ⇒ 1 = height/40 ⇒ height = 40 m
Q16
From the top of a tower 30 m high, the angle of depression of the top and bottom of a pole are 30° and 45°. Find the height of the pole.
  • A 26.08 m
  • B 20.8 m
  • C 25 m
  • D 28 m
Answer: Option B
Explanation: Let pole height = h. Using tan relations: distance = 30 (from 45°), then 30 - h = 30 × tan 30° ⇒ h ≈ 20.8 m
Q17
A man on top of a tower finds the angle of depression of a car as 30°. The car is 80 m from the base. Find the tower's height.
  • A 46.18 m
  • B 40 m
  • C 32 m
  • D 50 m
Answer: Option A
Explanation: tan 30° = height / base ⇒ 1/√3 = height/80 ⇒ height ≈ 46.18 m
Q18
A tree 10 m high casts a shadow 10√3 m long. Find the elevation angle of the sun.
  • A 45°
  • B 30°
  • C 60°
  • D 50°
Answer: Option B
Explanation: tan θ = height / shadow = 10/(10√3) = 1/√3 ⇒ θ = 30°
Q19
The top of one tower is 50 m higher than another. From a point between them, the angles of elevation of their tops are 30° and 60° respectively. Find the distance between the towers.
  • A 57.7 m
  • B 100 m
  • C 86.6 m
  • D 50 m
Answer: Option C
Explanation: Let heights be h and h+50. Using tan relations and solving gives distance between towers ≈ 86.6 m
Q20
The angle of elevation of a tower from a point on the same level is 45°. On advancing 30 m toward it, the angle changes to 60°. Find the tower's height.
  • A 30 m
  • B 20√3 m
  • C 25.98 m
  • D 15√3 m
Answer: Option C
Explanation: Let initial distance be x. Then tan 45° = h/x and tan 60° = h/(x-30). Solving gives h ≈ 25.98 m

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