Heights & Distances (ఎత్తు మరియుదూరం )

Heights & Distances (ఎత్తు మరియుదూరం )

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Heights & Distances (ఎత్తు మరియుదూరం)

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1. θ₁ and θ₂ are angles of elevation from two ends of horizontal rod of length L to top of vertical pole. Pole height h = L/(cotθ₁+cotθ₂)... verify.
Pole h, rod L. Elevations θ₁,θ₂ from rod ends to pole top. Verify h=L/(cotθ₁+cotθ₂).

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2. Cliff 150 m high. Two ships on sea, same side. Angles of depression 30° and 60°. Distance between ships?
Cliff=150m. Depressions 30° and 60° (same side). Ship distance?

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3. Shadow of pole at 3 PM = 2× shadow at noon. If noon elevation = 60°, find 3 PM elevation.
Noon elevation=60°, shadow doubles at 3PM. Find 3PM elevation.

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4. Tower and building same height h. Tower నుండి building top elevation = 30°, building నుండి tower top elevation = 60°. Distance between them?
Same height h. From tower: elevation to building top=30°. Building to tower top=60°. Distance?

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5. Two observers at A and B (3 km apart) simultaneously observe balloon. Elevations 60° and 30°. Balloon between them. Height?
A,B: 3km apart. Elevations 60°, 30° to balloon between them. Height?

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6. Angles of elevation and depression of top and bottom of tower from top of hill 200m high = 30° and 60°. Tower height?
Hill=200m. Elevation to tower top=30°, depression to tower base=60°. Tower?

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7. Person walks 20 m towards tower. Elevation changes from 30° to 45°. Tower height?
Elevation changes 30° to 45° walking 20m closer. Height?

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8. Isosceles triangle base 2a. From apex, angle of elevation of flag at midpoint of base = α. From base vertex, elevation = β. Flag height?
Isosceles triangle, apex to flag=α, vertex to flag=β. Flag height?

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9. Horizontal observer sees bird flying north at speed v m/s, constant height h. At t=0: elevation=α. At t=T: elevation=β. Find h in terms of v,T,α,β.
Bird flies north at v m/s, height h. Elevations α at t=0, β at t=T. Express h.

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10. Three collinear points A,B,C on ground. Tower at D (not on ABC line). From A,B,C: elevations 60°,45°,30°. AB=BC=100m. Tower height?
A,B,C collinear, AB=BC=100m. Elevations 60°,45°,30°. Height?

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11. River width అంటే bank నుండి directly cross bank కి distance. Bank నుండి tree top = 60°. Tree height = 20 m. River width?
Bank to opposite tree, elevation=60°, tree=20m. River width?

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12. Boy 150 cm tall. His shadow 2√3 m long. Angle of elevation of sun?
Boy=150cm, shadow=2√3 m. Sun angle?

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13. Tower AB = 30 m. Point C నుండి A మరియు B కి elevations 30° మరియు 45°. AC = ?
Tower AB=30m. Elevations to A and B from C: 30° and 45°. Find AC.

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14. Tower T. From A elevation = 45°. B is 30 m closer. Elevation from B = 60°. AT = ?
Elevation 45° at A, 60° at B (30m closer). Find AT (distance from A to tower).

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15. Prove: If α+β=90°, then pole of height h casts shadow h tanα on a slope that makes angle β with horizontal.
Prove shadow formula on inclined slope.

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16. Man 1.8 m tall, 10√3 m from lamp post. Shadow = 6 m. Lamp height?
Man=1.8m, 10√3 m from lamp, shadow=6m. Lamp height?

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17. Star at elevation θ. Earth radius R. Observer height h. True altitude?
Observer at height h (above sea level), sees star at elevation θ. Atmospheric refraction ignored. True altitude from center?

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18. Two towers A and B. Observer at C on ground. Elevations: A=60°, B=45°. AC=30m, BC=40m, ∠ACB=90°. Height of taller tower?
AC=30, BC=40, ∠ACB=90°. Elevations 60°,45°. Taller tower?

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19. Two towers of same height on either side of road (width 80 m). From middle of road, elevations = 30° each. Height?
Road=80m. From middle: elevations=30° to both towers. Height?

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20. Elevation to top of mountain = 45°. Walking 1 km towards it, elevation = 60°. Mountain height?
Elevation: 45° then 60° after walking 1km. Mountain height?

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21. From top of hill 200 m high, angles of depression of two boats = 30° and 45°. Distance between boats?
Hill=200m. Depressions to boats: 30° and 45°. Distance between boats?

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22. Gradient of hill = 1:5 (rise:run). Angle of inclination? (tanθ=1/5)
Hill gradient 1:5. Find angle of inclination.

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23. Square base tower. From midpoint of side, elevation = α. From corner, elevation = β. If α > β, prove tanβ = tanα/√2... adjust.
From midpoint of base side: elevation=α. From corner: elevation=β. tanα/tanβ=?

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24. Navigation: Ship at S sees lighthouse L at bearing N30°E, elevation 20°. After sailing 5 km north, sees L at bearing N60°E, elevation 30°. Height of L?
Navigation problem. Lighthouse height?

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25. Tower height h. From point A (north): elevation = α. From B (east): elevation = β. A and B same distance from tower. AB = ?
Tower h, A north, B east, same distance d from tower. Elevations α,β. AB?

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26. Aeroplane at 3000 m. Angles of depression of two ships (same side) = 60° and 45°. Distance between ships?
Height=3000m. Depressions 60° and 45°. Ship distance?

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27. Ladder of length 10 m makes 60° with ground. Height reached on wall?
Ladder=10m, angle with ground=60°. Height on wall?

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28. Angle of elevation to sun changes from α to β. In this time, shadow length changes from a to b. Sun is at infinity. Prove: (b-a) = h(cotβ-cotα).
Prove shadow change formula.

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29. Ship A at 30° elevation, Ship B at 45° elevation from lighthouse top 100 m. Same line, opposite sides. Distance AB?
Lighthouse=100m. Ship A=30°, Ship B=45° (opposite sides). Distance?

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30. Window is 8 m above ground. Angle of depression to car = 60°. Distance of car from building?
Window 8m high, depression to car=60°. Car distance?

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31. Optimal angle of projection for maximum range on inclined plane (angle α to horizontal) = 45°+α/2. Derive briefly.
Optimal angle for max range on slope α?

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32. Tower foot నుండి 100 m దూరంలో angle of elevation = 30°. Tower height?
Distance=100m, elevation=30°. Tower height?

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33. Standing on bridge, angle of depression to boat upstream = 30°, downstream = 60°. Bridge height = 10 m. Distance between boats?
Bridge=10m. Depressions upstream=30°, downstream=60°. Boat distance?

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34. Height = 15 m, angle of elevation = 30°. Distance from base?
Height=15m, elevation=30°. Distance?

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35. Kite is flying at height 60 m. String makes 60° with ground. String length?
Kite height=60m, string angle=60°. String length?

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36. Object moves horizontally. Elevation changes from 45° to 30° in 3 seconds at 50 m/s. Initial distance?
Object moves at 50m/s. Elevation 45°→30° in 3 sec. Height?

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37. From top of tower h m, elevation of hilltop = 60°, depression of hill foot = 30°. Hill height?
Tower=h. From top: hilltop elevation=60°, hill foot depression=30°. Hill height?

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38. Sun's angle of elevation = 45°. Pole height = 12 m. Shadow length?
Sun elevation=45°, pole=12m. Shadow?

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39. From top of lighthouse 100 m high, depression of ship = 30°. Ship approaching at 25 m/s. Time to reach base?
Lighthouse=100m, depression=30°, ship speed=25m/s. Time?

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40. Angle of elevation of sun changes from 30° to 60° over 2 hours. If shadow at 30° is 60m, shadow at 60°? Rate of change of shadow length?
Shadow at 30°=60m. Find shadow at 60° and rate of change.

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41. Pole shadow = height × √3. Angle of elevation of sun?
Shadow = height×√3. Sun's angle?

42 / 89

42. Pole at corner of square field side a. From opposite corner, elevation = 60°. Height?
Square side=a. Pole at corner, elevation from opposite corner=60°. Height?

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43. Man 1.6 m tall walks away from lamp (5 m). Shadow lengthens at 2/3 m per second when man walks at 1 m/s. Verify.
Lamp=5m, man=1.6m, walks at 1m/s. Shadow length rate = 2/3 m/s?

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44. Person stands 40 m from building. Elevation to top = 60°, to bottom of flag = 45°. Flag height?
Building top elevation=60°, flag bottom=45°, distance=40m. Flag height?

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45. Ship is 100√3 m away from lighthouse. Angle of elevation of top = 30°. Height of lighthouse?
Ship 100√3 m away, elevation=30°. Lighthouse height?

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46. Horizontal distance = 20 m, angle of elevation = 45°. Height?
Distance=20m, angle=45°. Height?

47 / 89

47. Balloon rises vertically from A. B is 100 m from A. From B, elevation changes from 30° to 60°. Distance balloon rose?
Balloon rises from A. B is 100m away. Elevation 30°→60°. Distance risen?

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48. From vertex of equilateral triangle, angle of elevation of top of vertical pole at centroid = 60°. Triangle side = a. Pole height?
Equilateral triangle side=a. Pole at centroid, elevation from vertex=60°. Height?

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49. Tower foot నుండి 30 m దూరంలో elevation = 60°. Tower నుండి 60 m దూరంలో elevation = ?
Elevation=60° at 30m. Find elevation at 60m.

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50. Tower casts 40 m shadow when sun elevation = 30°. Tower height?
Shadow=40m, elevation=30°. Height?

51 / 89

51. Two towers on opposite banks of river. Angles of elevation from midpoint = α,β. From one bank end = γ to closer tower. River width = w. Tower heights?
River width w. From mid: elevations α,β. Find tower heights.

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52. Angle of elevation of cloud from point 60 m above lake = 30°. Angle of depression of its reflection = 60°. Height of cloud above lake?
60m above lake: cloud elevation=30°, reflection depression=60°. Cloud height?

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53. Elevation to top of hill from bottom of tower = 60°. Elevation to top of tower from bottom of hill = 30°. Tower = 50 m. Hill height and distance?
Tower=50m. Elevation hill→tower top=60°, tower→hill top=30°. Hill height?

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54. Tower leans at θ to vertical. From base: distance d, elevation α. From same side, distance 2d, elevation β. Tower height?
Leaning tower, angles α,β from d and 2d. Height in terms of d,α,β?

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55. Tower 60 m high. From bottom angle of elevation of cloud = 30°, reflection in lake = 60°. Cloud height above lake?
Tower=60m, elevation to cloud=30°, reflection=60°. Cloud height above lake?

56 / 89

56. Two buildings h₁ and h₂ apart by distance d. From bottom of h₂, elevation of top of h₁ = α. h₁ cosec α = ?
From bottom of h₂: elevation to top of h₁=α. Express h₁/sinα.

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57. Person sees top of building at 45° elevation. Walks 10m towards building, sees at 60°. Walks another 5m. What elevation now?
Elevation 45° then 60° after 10m. After 5m more, find elevation.

58 / 89

58. Parallax method: Star elevation changes by p (parallax) when observer moves d km. Distance to star?
Parallax angle p (radians), baseline d. Star distance D?

59 / 89

59. Two persons 1.6 m tall each. One on top of hill 40 m high, sees other at 30° depression. Distance?
Hill=40m. Person on top sees other person (1.6m tall) at 30° depression. Distance?

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60. Angle of elevation = 45°, height = 50 m. Horizontal distance?
Elevation=45°, height=50m. Distance?

61 / 89

61. Sine rule in height problems: non-right triangles. Tower AB, observer C. ∠ACB=θ, BC=a, ∠BAC=α. AB=?
In triangle ABC: ∠ACB=θ, BC=a, ∠BAC=α. Find AB using sine rule.

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62. Angle of depression from top of cliff to boat = 45°. Cliff height = 50 m. Distance of boat?
Depression=45°, cliff=50m. Boat distance?

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63. Person at sea level sees cliff top at 60°, cliff foot at 30° (cliff on hill). If sea-to-hill base = 400 m, cliff height?
Sea level: cliff top=60°, hill base=30°. Sea to hill=400m. Cliff height?

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64. Two poles heights 6 m and 11 m, 12 m apart. Wire joining tops. Wire length?
Poles 6m and 11m, 12m apart. Wire between tops?

65 / 89

65. Angle of elevation of plane from two points A and B (1 km apart, same line) = 60° and 30°. Height of plane?
Plane elevations 60° and 30° from A,B (1km apart). Height?

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66. From top of cliff h m, elevation of top of lighthouse = θ₁, depression of its foot = θ₂. Lighthouse height?
Cliff=h. Elevation to lighthouse top=θ₁, depression to foot=θ₂. Lighthouse height?

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67. Tree of height 10√3 m casts shadow 10 m. Angle of elevation of sun?
Tree=10√3 m, shadow=10m. Sun's elevation?

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68. Tower is observed from two points A and B on same horizontal plane. A, B on same line. Elevation from A = α, from B = β. AB = d. Tower height?
Elevations α,β from A,B (d apart, same line). Height?

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69. Angle of elevation of top of tower = 30°. Elevation of point halfway up tower = β. Prove tanβ = 1/√3 × 2.
From same point: elevation to top=30°, to halfway point=β. Prove tanβ.

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70. Flagpole height = h m. Observer at angle 30° sees top. Distance = ?
Flagpole=h m, elevation=30°. Distance in terms of h?

71 / 89

71. Flag on top of tower. From 9 m away: elevation to flag bottom = 30°, flag top = 60°. Flag length?
9m away: flag bottom elevation=30°, top=60°. Flag length?

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72. Angle of depression అంటే ఏమిటి?
What is angle of depression?

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73. Angle of elevation అంటే ఏమిటి?
What is angle of elevation?

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74. Flagpole on top of building. From 30 m away: elevation to building top = 45°, flagpole top = 60°. Flagpole height?
Distance=30m. Building top=45°, flagpole top=60°. Flagpole height?

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75. 30° angle of elevation వద్ద shadow length = 30 m. Tower height?
Angle=30°, shadow=30m. Tower height?

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76. Angle subtended by tower at base of flagpole = 45°. Angle subtended by flagpole at base of tower = 30°. Heights 20m and h. Find h.
Tower=20m subtends 45° at flagpole base. Flagpole subtends 30° at tower base. Flagpole=h?

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77. From top of 30 m tower, angle of depression of car = 30°. Car distance from tower base?
Tower=30m, depression=30°. Car distance?

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78. Building height = 20 m. Shadow at noon = 20/√3 m. Sun's elevation?
Building=20m, shadow=20/√3 m. Sun angle?

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79. Tower A, B same base. From C: elevation to A = 30°, elevation to B = 60°. From D midpoint of AB: elevation to C = 45°. CD = AB/2. Relation?
Complex multi-point problem. Standard result?

80 / 89

80. Aircraft flying at 1000 m height. Angle of depression to airport = 30°. Horizontal distance?
Aircraft height=1000m, depression=30°. Horizontal distance?

81 / 89

81. Tower height = 10 m, horizontal distance = 10 m. Angle of elevation = ?
Tower height=10m, distance=10m. Angle of elevation?

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82. Angle of elevation of top of tower from foot of pole = 60°. Angle of elevation of top of pole from foot of tower = 30°. Pole = 50 m. Tower height?
Pole=50m. Elevation pole→tower=60°, tower→pole=30°. Tower height?

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83. Angle of elevation of top of tree = 60°. Distance from tree = 20 m. Height?
Elevation=60°, distance=20m. Height of tree?

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84. Elevation to top of tower = 30°. Walk x m nearer, elevation = 60°. Height of tower in terms of x?
Elevation 30°→60° after walking x m. Tower height?

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85. Theodolite at O measures elevation to tower top T = θ₁ from ground. From point P (directly below T), elevation to T = 90°. OP = d. OT = ?
O on ground, T = tower top. From O: elevation=θ₁. OP=d (below T). Find OT.

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86. Pole height = 6 m, shadow = 6 m. Elevation angle of sun?
Pole=6m, shadow=6m. Sun angle?

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87. From bank of river, elevation of tree on opposite bank = 60°. 4 m back, elevation = 30°. River width and tree height?
Elevations 60° and 30° from two points 4m apart. Width and height?

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88. Helicopter at height 500 m. Angle of depression of two cars on road = 60° and 30° (same side). Distance between cars?
Height=500m. Depressions 60° and 30°. Car distance?

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89. From ground, angle of elevation of cloud = 30°. Height of cloud = 100 m. Horizontal distance?
Cloud elevation=30°, height=100m. Distance?

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