Height and Distanceऊँचाई और दूरी questions and answersप्रश्न और उत्तर

How height and distance worksऊँचाई और दूरी को समझिए

The angle of elevation is measured up from the horizontal at the observer’s eye; the angle of depression is measured down from the horizontal at the top. A horizontal line at the top is parallel to the ground, so the angle of depression from the top equals the angle of elevation from the point below.

In each right triangle the height is opposite the angle and the ground distance is next to it, so height = distance × tan θ and distance = height × cot θ. Use sine or cosine only when the line of sight itself (the slant side) is given or asked.

When two observations are made, write each distance in terms of the height and use the fact that links them: the walk between two points on the same side is the difference of the distances, while two objects on opposite sides are separated by the sum.

Each trick says when to use it, why it works and when it fails. Formula tricks are checked by code over their whole range.हर ट्रिक बताती है कि उसे कब इस्तेमाल करें, वह क्यों काम करती है और कब नहीं। सूत्र वाली ट्रिक्स को कोड से उनकी पूरी रेंज पर जाँचा गया है।

Answers and full solutions are shown. Hide them to test yourself first.उत्तर और पूरे हल दिखाए गए हैं। पहले खुद को परखने के लिए इन्हें छिपाइए।

1. Example 1उदाहरण 1 · Heights and distances with 30°, 45° and 60° · Easyआसान

   Rohan stands on level ground in Surat, 60 m from the foot of a clock tower. The angle of elevation of its top is 60°. How tall is it?

   1. 20√3 m
   2. 30 m
   3. 30√3 m
   4. 60√3 m
   Answer:उत्तर: 60√3 m

   1. tan 60° = height/60.
   2. Height = 60 × √3 = 60√3 m.

2. Example 2उदाहरण 2 · Heights and distances with 30°, 45° and 60° · Mediumमध्यम

   Saanvi looks at the top of an office building in Surat and finds the angle of elevation is 30°. After walking 60 m straight towards it, the angle becomes 60°. How far is Saanvi now from its foot?

   1. 60 m
   2. 30√3 m
   3. 90 m
   4. 30 m
   Answer:उत्तर: 30 m

   1. Far point: distance = h cot 30° = √3·h. Near point: distance = h cot 60° = 1/√3·h.
   2. √3·h − 1/√3·h = 60 ⇒ h = 30√3 m.
   3. Distance from the near point = h × 1/√3 = 30 m.

3. Example 3उदाहरण 3 · Heights and distances with 30°, 45° and 60° · Hardकठिन

   From the top of a water tank 36 m high, two cars on opposite sides of it, on a straight road through its foot, are seen at angles of depression of 45° and 60°. How far apart are the cars?

   1. 36(√3 + 1) m
   2. 12(3 + √3) m
   3. 12(3 − √3) m
   4. 36 m
   Answer:उत्तर: 12(3 + √3) m

   1. First car: 36 × cot 45° = 36 m.
   2. Second car: 36 × cot 60° = 12√3 m.
   3. Opposite sides, so the gap is the sum: 12(3 + √3) m.

4. Example 4उदाहरण 4 · Heights and distances with 30°, 45° and 60° · Mediumमध्यम

   Rohan looks at the top of a temple gopuram in Jaipur and finds the angle of elevation is 30°. After walking 54 m straight towards it, the angle becomes 45°. How far is Rohan now from its foot?

   1. 27(3 + √3) m
   2. 54 m
   3. 27(√3 + 1) m
   4. 27√3 m
   Answer:उत्तर: 27(√3 + 1) m

   1. Far point: distance = h cot 30° = √3·h. Near point: distance = h cot 45° = 1·h.
   2. √3·h − 1·h = 54 ⇒ h = 27(√3 + 1) m.
   3. Distance from the near point = h × 1 = 27(√3 + 1) m.

5. Example 5उदाहरण 5 · Heights and distances with 30°, 45° and 60° · Mediumमध्यम

   Lakshmi looks at the top of a temple gopuram in Indore and finds the angle of elevation is 30°. After walking 84 m straight towards it, the angle becomes 60°. How tall is it?

   1. 56√3 m
   2. 84√3 m
   3. 21√3 m
   4. 42√3 m
   Answer:उत्तर: 42√3 m

   1. Far point: distance = h cot 30° = √3·h. Near point: distance = h cot 60° = 1/√3·h.
   2. √3·h − 1/√3·h = 84 ⇒ h = 42√3 m.
   3. Distance from the near point = h × 1/√3 = 42 m.

6. Example 6उदाहरण 6 · Heights and distances with 30°, 45° and 60° · Mediumमध्यम

   Vihaan looks at the top of a temple gopuram in Surat and finds the angle of elevation is 30°. After walking 48 m straight towards it, the angle becomes 45°. How far is Vihaan now from its foot?

   1. 48 m
   2. 24(√3 + 1) m
   3. 24(3 + √3) m
   4. 24√3 m
   Answer:उत्तर: 24(√3 + 1) m

   1. Far point: distance = h cot 30° = √3·h. Near point: distance = h cot 45° = 1·h.
   2. √3·h − 1·h = 48 ⇒ h = 24(√3 + 1) m.
   3. Distance from the near point = h × 1 = 24(√3 + 1) m.

Common mistakesआम गलतियाँ

• Swapping tan and cot, which gives a height of d/√3 instead of d√3.
• Using sin θ when the known side is the ground distance, not the line of sight.
• Adding distances for points on the same side, or subtracting for points on opposite sides.

Frequently asked questionsअक्सर पूछे जाने वाले प्रश्न