Page 1 : Line of sight

Page 2 : Solution:, , Iet AB be the tower and C be the point of observation, , BC 120 m, Since, angle of elevation is 30°, , LACB =30°, 120 m, , AB, tan 30°, , InA ABC, 190, , AB 120 x, , 69-28 m, , 3, , Ans., , 3 A guard observes an enemy boat, from an observation tower at a height of, , 180 m above sea level, to be at an angle of depression of 29., () Calculate, to the nearest metre, the distance of the boat from the foot of, the observation tower., , (i) After some time, it is observed that the boat is 200 m from the foot of the, , observation tower. Calculate the new angle of depression., Solution:, , ) Let P be the boat and AB be the observation tower. In right triangle ABP,, AB, , 29, , tan 29° =, , A, , PB, E, , 180, , 0-5543 =F, , PB, , 180, , PB 0-5543, , =, , 325, , P, , (App.), , m, , 29, , DB, , Ans., , Alternative Method:, In A ABP, ZAPB + ZPAB, , = 90°, , ZPAB = 90° -, , 29° = 61°, , BP, = tan 61°, , AB, BP, , 180 x tan 61°, , =, , 180 x 1-804 =, , 325 m (App.), , Ans., , () Let Q be the new position of the boat and 6 be the new angle of depression, AB, , tan6= QB, 180, , = 0-9000, , 200, =, , 0, , 4, , tan 41°, , 59, , P, , 200m, , 41°59, , Ans., , IWo people standing on the same side of a tower in a straight line with it,, , measure the angles of elevation of the top of the tower as 25° and 50, , espectively. If the height of the tower is 70 m, find the distance between the, two people., , 335

Page 3 : Solution, According to the given statement, the figure, will be as shown alongside in which PQ is the, , tower so PQ = 70 m; and A and B be the, pOsitions of two people, such that ZPAQ = 25, and PBQ = 50°, , In A PAQ, , tan 25, , =, , In A PBQ, , tan, , 25, , PQ, AQ, , 70, , U003, , R, , 50, , 70, , AO, , ie. AQ = .4663 m = 150-118 m, , PQ, , 50, , =, , BO, 70 Le., , 70 m =58-735 m, 1-1918, , 1-1918 = RO, , 'The distance between the, =, , AB, , people, , two, , =AQ, , = (150-118, , BQ, , -, , 58-735) m, , =91-38 m, , Ans., , Alternative method, In A PA,, , ZA = 25°, LAPQ = 90°, tan 65, , 65, , 25° = 65°, , 40, , AQ, , =, , A, , PQ, , i.e., , AQ, 2-1445 70, , AQ, , In A PBQ,, , PBQ, , B P Q =90°, , = 50°, , =, , 2-1445, , x, , 50, , 25, 70, , m =150-115, , m, , 50° = 40, , BQ, , tan 40°= PO, i.e., , 0-8391, , =, , BQ, , BQ, , 70, , T h e distance between the two, AB, , = AQ, , -, , BQ =(150-115, , -, , 08391, , =, , x, , 70, , m =58-737, , m, , people, , 58-737), , m, , =, , 91-38, , m, , Ans., , EXERCISE 22(A), , The height, , of, , is, , N3 times the length, of its shadow. Find the angle of elevation of, a, , tree, , |3.A, , the sun., , 2., , The angle of elevation of the top of a tower,, from a point on the ground and at a distance, of 160 m from its foot, is found to be 60°., Find the height of the, , tower., , ladder is placed, along a wall such that its, upper end is resting against a vertical, wall. Th, foot of the ladder is, 2-4 m from the wall and, the ladder is, , ground., reaches., , making an angle of 68° with the, height, upto which the ladder, , Find the, , A. Two persons are standing on the opposite sides, , 336

Page 4 : af a, , observe the, , They, , tower., , tower is 45 m. Calculate, ) the height of the tower, , angles of, , be, top, elevation, Find, the distance, 30 and 38 respectively., the, of, the, if, tower is, height, between them,, of the tower to, , of the, , Gi) the length of the shadow of the same, tower, when the sun's altitude is, (b) 60, (a) 45, , 50 m., a, is attached to, string Find the length, 5. Aof kite, the string, when the height of the kite is, , 10, Two vertical poles are on either side of a road., A 30 m long ladder is placed betwéen the two, , angle 30 with, , poles. When the ladder rests against one pole,, it makes angle 32 24 with the pole and when, it is turned to rest against another pole, it, , makes, 60 m and the string, the ground, , (6A boy,, , 1-6, , m, , and observes, of the, , tower to, , height of the, , away from a tower, angle of elevation of the top, 60. Find the, be (1) 45 ), , tall, is 20, the, , an, , tower, , m, , makes angle 32 24' with the road. Calculate, the width of the road., , 11. Two climbers are at points A and B on a, vertical cliff face. To an observer C, 40 m, , in each case., , ot a tree, broken over by the, 7. The upper part, wind, makes an angle of 45 with the ground:, and the distance from the root to the point, where the top of the tree touches the ground,, of the tree, is 15 m. What was the height, , before it, , was, , from the foot of the cliff, on the level ground,, A is at an elevation of 48° and B of 57. What, is the distance between the climbers ?, , (12. Aman stands 9, , away from, , a, , flag-pole., , He, , observes that angle of elevation of the top of, the pole is 28° and the angle of depression of, , broken ?, , of elevation of the top of an, unfinished tower from a point at a distance of, , 8The angle, , the bottom of the pole is 13 Calculate the, height of the pole., , 80 m from its base is 30. How much higher, must the tower be raised so that its angle of, , 13. From the top of a cliff 92 m high. the angle, of depression of a buoy is 20°. Calculate,, , plevation at the same point may be 60°?, (9 At a particular time, when the sun's altitude, , to the nearest metre, the distance of the buoy, , is, 3 0 ° , the length of the shadow of a vertical, , 5, , m, , from the foot of the cliff., , The length of the shadow of a vertical tower on level ground increases by, , 10 m, when the altitude of the sun changes from 45° to 30. Calculate the, height of the tower, correct to two decimal places., , Solution:, Let AB be the tower. BC be its shadow when the, sun's altitude is 45°, i.e. ZACB = 45°., , When the sun's altitude changes to 30, the length, of the shadow is BD and 2ADB = 30°, , Clearly, DC = 10 m., , BL, , In triangle ABC,, tan 45° =, , AB, , BC, In triangle ABD,, tan 30° =, , Since, , AB, , 1 BC, , AB, , BD, , BD, , AB3, , BC=, , 30, , C-10 m-, , BC = AB, , AB, , BD -, , 45, , BD =AB 3, , DC, , AB = 10, , :BC = AB and BD =AB/3, , 337, , D

Page 5 : AB(3-1) = 10, 10, , AB, , -1 5-1, 10(1-732, , 1), , +, , 5x 2-732, 5x, 2-732, , =, , 3-1, 6, , 3+1, 3+1, , 10, , 13-66, , =, , m, , Ans., , An observer on the top of a cliff; 200 m above the, sea-level, observes the, angles of depression of the two ships to be 45 and 30, respectively. Find, the distance between the ships, if the ships are, (0 on the same side of the cliff, (i) on the opposite sides, of, the cliff., , Solution:, ) Ships on the same side, of the clif, In the figure, cliff AB, 200 m., , 458, , =, , C and D, , are, , the, , positions, , of, , ships., , two, , Required to find: Length of CD., In, right-angled triangle ABC,, tan 45= 200, , right-angled triangle ABD, , tan 30°, , 200, , BC, , BC, , 200m., , 200, , RD, , DBD 200 V3 m, , BD, , Distance between the, ships, i) When the, ships, of the cliff, , are on, , A man on the, top of, , 200, , =, , =, , CD, , =, , BD, , 1-732, , x, , m, , =, , 146-4, , ships, , a, , CD, , =, , BD, , =, , 346-4, , +, , 346-4, , m, , 45, =, , =, , m, , BC, , 346-4 m-200, the, opposite sides, , The distance, between the, , 7, , 45, , 200, , BC, , In, , DA30, , 45, , BC, m, , +200, , m, , =, , Ans., , 30, B, , 546-4, , m, , Ans., , vertical, , observation tower observes, uniform speed, a car, of depression tocoming directly towards it. If it, moving at a, 12, change, takes, reach the observation towerfrom? 30 to 45, how soon minutes for the angle, after this will the car, Solution, car, , Let AB be the, tower and, reaches the position D. C be the, Clearly, 2ACB 30° and, 2ADB, Let the, speed of the, , initial, , =, , observation tower., , position, , of the car., After 12, , =45, , car bex m/minute, 338, , and it will, taket, , minutes, , minutes,, , the, , reach, , the, , to

Page 6 : CD = 12r m, , Since, distance = time x speed), , and DB = xm, , In A ABD., , tan 45 =1h=tx, , 45, , In A ACB. 12x+ tx tan 30°, (Since h = t), , (12+ 1) x, , 2x, , 3r=12 +, , 45, , 3r-I =12, r(3-1), , 12, , ==12, , I= 3 - 1, , l6-39 minutes, , Ans., , 8The angle of elevation of a stationary cloud from a point 25 m above a lake is, 30 and the angle of depression of its reflection in the lake is 60. Whatis, the height of the cloud above that lake-level ?, , Solution, Let AB be the surface of the lake, P be the, point of observation which is 25 m above the, lake ie PQ = 25 m., , h, hm, 25 m 0, , IfCbe the cloud andD be its reflection in, the lake then according to the properties of, reflection. the height of cloud C above the lake, level is equal to the depth of its image D below, , h, , the lake level ie. CF = DF = h m let), , As is clear from the, , DPE = 60°,, and,, , Clearly, , figure,, , 2CPE, , =, , 30, , CE = CF - EF = (h -25) m, DE = DF + EF = (h + 25) m, , CE, tan 30= PE, , [In A CPE], , and, , And, , PE, , tan 60=DE, , [In A DPE, , PE, , 3, , = +25, , (h-25), , nd PE=, , (, , From I and II, we have, , 3 (h-25), , =, , 3, , 3h 75 =h+ 25, , ie, , 3h 25) =h + 25, , ie, , h =50, , The height of the cloud above the lake-level x = 50 m, 339, , Ans., , a

Page 7 : 9, , From a point on the ground. the ange of elevation, of the tog af a wert, tower is found to be such that its tangent is, On, tower, the tangent of the new ange of elevation walking 50 mtowarts e, of the top of the, found to be, , toweris, , Find the height of the tower, , Solation, Let AB be the tower whose beight is à mP, point of observation such that if the, angle of elevation of the top of the tower at, point P is a then tan a= i, be the first, , Let, , Q be the other point of, obtained on mowing m uowads the, (Le. PQ= 50 m) such that if theobservation, wer, angle of elevation of the op of50, the, tower, an poime Qs, B. then tan B=, Assume BQ, , to, , be, , In A APB, , A, tan a=a, , 50, , 5h= 3x, , AB, , In A AQB:, , On, , xm, , 150, , tanB- QB, , solving I and Il we get: h, The height of the tower, , 5h =4x, 120, , =, , =, , 120, , 10 Avertical pole and a, vertical tower are on the same lesel, of the pole the ange of, gound. Fram the tup, of the top of the tower is, of depression of the footelevation, 60 and the ange, of the tower is 30 Find, the height of the pole is 20m, the height of the twer i, , Solution, According to the given statement, the, where BC is the pole and, AE is the tower, , diagram wibe as shown om the next, page, , Clearly, BC 20 m= DE. ZABD 60, Let AD =xm and so AE (z 20) m, In right-angled A, ABD., =, , =, , =, , tan 60, In, , = AD, BD, , right-angled, , A, , EBD., , nd EBD, , +, , BD-, , DE, , tan 30= BD, , 340, , =, , 30

Page 8 : BD =, , 203, , ...I, , From equations I and II, we get, , 520/3, X = 60, , D, , Height of the tower = AE = (x +20)m, = (60 + 20) m, , = 80 m, , 20 1, , 20 m, , Ans., , C, , EXERCISE 22(B), 1. In the figure, given below, it is given that AB is, perpendicular to BD and is of length X metres., DC=30 m, ZADB =30° and ACB 45., , 22, , =, , Without using tables, find X., , X, , E, , C47, , 15 m, HD, , D, , 30 m, , T h e angle of elevation of the top of a tower, is observed to be 60°. At a point, 30 m, vertically above the first point of observation,, the elevation is found to be 45°. Find:, ) the height of the tower,, i) its horizontal distance from the points of, , 45, , 30, C, , 2. Find the height of a tree when it is found that, on walking away from it 20 m, in a horizontal, line through its base, the elevation of its top, changes from 60° to 30°., , observation., , 3Find, . the height of a building, when it is found, , 8. From the top of a cliff, 60 metres high, the, angles of depression of the top and bottom of, , that on walking towards it 40 m in a horizontal, , a tower are observed to be 30° and 60°. Find, , line through its base the angular elevation of, , its top changes from, , 30°, , 4. From the top of a light, , to, , the height of the tower, , 45°, , house 100, , m, , high,, , 9. A man on a cliff observes a boat, at an angle, of depression 30°, which is sailing towards the, , the, , angles of depression of two ships are observed, as 48 and 36 respectively. Find the distance, , shore to the point immediately beneath him., Three minutes later, the angle of depression of, , between the two ships (in the nearest metre), if, , ), , the boat is found to be 60°. Assuming that the, , boat sails at a uniform speed, determine, , the ships are on the same side of the light, , 1) how, , house,, , theshore ?, , (11) the ships are on the opposite sides of the, , light house., , 2010], , .Two pillars of equal heights stand on either, , much more time it will take to reach, , i), , the, , H, , of the boat in metre per second,, height of the cliff is 500 m., , speed, , if the, , SIde of a roadway, which is 150 m wide. At a, , 10. A man in a boat rowing away from a lighthouse, , point in the roadway between the pillars the, of the tops of the pillars are 60° and, , of elevation of the top of the lighthouse from, 60° to 45°. Find the speed of the boat., , elevations, , 30, find the height of the pillars and the, , position of the point., , 150 m high, takes 2 minutes to change the angle, , bank of a river, person standing on the, of the top, observes that the angle of elevation, bank is 60., of a tree standing on the opposite, , I1. A, , 6. From the figure, given below, calculate the, , length of CD., 341

Page 9 : When he moves 40 m away from the bank,, he finds the angle of elevation to be, 30°, , Find:, ), , the height of the tree, correct to 2, decimal places,, (ii) the width of the river., , 12. The horizontal distance between, , two towers, 1s 75 m and the angular depression of the top, , when it was 45. Prove that the height of the, , tower is y(V3+ 1) metres., 14. An aeroplane flying horizontally 1 km above, the ground and going away from the observer, , is observed at an elevation of 60°. After 10, seconds, its elevation is observed to be 30°;, , find the uniform speed of the aeroplane in km, p e r hour., , ot the first tower as seen from the top of the15. From the top of a hill, the angles of depression, , second, which is 160 m high, is 45. Find the, height of the first tower., 13. The length of the shadow of, on, , level, , plane, , a, , tower, , is found to be, , standing, , 2y metres, longer when the sun's altitude is 30° than, , of two consecutive kilometre stones, due east,, , are found to be 30° and 45° respectively. Find, the distances of the two stones from the foot, of the hil1, , EXERCISE 22(C), 1. Find AD., , calculate, , i), , ), , the length of AB;, , i) the distance of AB from the centre C., , B32, , E, , 6. At, , a, , point, , level, , on, , ground,, , the, , angle, , of, , elevation of a vertical tower is found to be such, , 2m, 0, , that its tangent, , ), , is. On walking 192 metres, towards the tower; the tangent of the, angle is, , 30, m, , found to be, , AB, , D, , 3, , Find the height of the tower., , 7. A vertical tower stands on horizontal, is surmounted by a vertical, , plane and, of height, plane, the angle of, , 2. In the following, diagram,, AB is a floor-board;, , flagstaff, , h metre. At a, point on the, elevation of the bottom of the, flagstaff is a, and that of the top of, is, flagstaff B. Prove that, the height of the tower is, , PQRS is a cubical box, with each edge = 1m, and 2B = 60°. Calculate, the length of the board, , Q, , AB, 60, , tan, , B, , htan a, , - tan a, , C, , 8. With reference to the, given, figure, a man stands on the, ground at point A,, which is on the, , 3. Calculate BC., , 42°, , horizontal, , same, , plane, , as, , B,, , D, , the foot, , of the vertical, , pole, , B, , BC. The height of, 4. Calculate AB., , the, the, , 6m, , 5, , m, , pole, , is 10, , m., , The man's eye is 2 m above, angle of elevation, , ground. He observes the, of C, the, top of the pole,, , 7, , 30, , =, , 5. The radius of a circle is given as 15 cm and, chord AB subtends an angle of 131° at the, centre C of the circle. Using, , trigonometry,, , 342, , Calculate, , ), , as, , the, , x, where, , tan, , distance AB in, , metres; (i) angle of elevation of the top of the, , pole when he is standing 15 metres from the, , pole., , Give your answer to the nearest, , degree

Page 10 : of, , 0. The angles, , elevation of, , the, , top of, , a, , 16. A man observes the angle of elevation of the, , tower, , be 30°. He walks towards, it in a horizontal line through its base. On, , ground at distances a, base, of the tower and, the, from, and b metres, line with it are, same straight, in the, Provo that the height of the, complementary., tower, , 10., , From, , elevation, angle of, , to the nearest metre., , of the, , top C of a, , x, , and, , where tan, , is x,, , 17. As observed from the top of a 80 m tall, lighthouse, the angles of depression of two, house in, ships, on the same side of the light, and 40°, 30°, are, with, its, base,, horizontal line, , =, , respectively. Find the distance between the, the nearest, ships. Give your answer correct to, 2012], two, , depression of, the angle of, of the, the foot D, tower, , is, , 18. In the given figure,, , (See A, , tan y=F, , from, , figure)., , Calculate the, , CD of the, , /metre., , where, , y,, , the given, , tower, , in, , D, , B, , height, , tower, , that the, tower,, , and the, the, , angle, , as seen, , 60 respectively. Find:, , ), , such a way, the, of, top of the, of elevation, is 60°, from the foot of the pole,, elevation of the top of, , pole, , seen, , altitude of 250 m,, 19. An aeroplane, at an, of two boats, observes the angles of depression, be 45° and, to, river, of a, on the opposite banks, on the, are, If the boats, 60 respectively., find the width, opposite sides of the aeroplane,, correct to the, of the river. Write the answer, nearest whole number., , ground in, , from the foot of the, , distance, , pole and, , A, , between, , is 120, angle of, , two towers, , tower, , The, , m., , elevation of the top, E, of, angle, and, depression of thee, , a, , vertical, , tower are on, , the, , bottom of the first, , the, , way that from, of the, top of the pole the angle of elevation, of, top of the tower is 60° and the angle, 30., is, tower, the, depression of the bottom of, level, , horizontal, , 20. The, , and the tower., Same, , B, , horizontal, , the, , a(i) the height of the lamp post., , tower., is 30°. Find, (i) the height of the, the pole, (1) the horizontal distance between, , 14. A vertical, , D, , distance between AB and CD., , angle of, as, , 60, , CD are, observed to be 30° and, , ?, , the same level, , 30, , ofa, , lamp post, , the bank of a river observes, 12. A man standing on, elevation of a tree on the, that the angle of, When he moves 50 m away, opposite bank is 60"., of elevation to, from the bank, he finds the angle, width of the river and, be 30°. Calculate: () the, (i) the height of the tree., and a vertical tower, 13. A 20 m high vertical pole, are on, , top, , depression of the top, and bottom of a vertical, , metres., , tower, , the foot of the, , the, , building AB =60 m, high, the angles of, , A man standing, tower is 20 m high., 11. A vertical, from the tower knows that the, at some distance, elevation of the top of, cosine of the angle of, from, is 0-53. How far is he standing, , the, , to, , to 60. Find the height of the building correct, , A, TO m above the ground the, , a window, , building, , a, , cOvering 60 m, the angle of elevation changes, , metre., , is vab, , tower, , top of, , on the, , points, , two, , from, , ground in such, , a, , tower as observed, , from the top of the, second, , 24, , towers., , respectively. Find, answer, , Give your, , height of the two, correct to 3 significant figures., , (i) the height of the, , 21. The, , pole, if the height of the tower is 75 m., , is 30° and, , the, , Find: ) the height of the tower, if the height, of the pole is 20 m;, , tower, , D, , BA, , 15. From a point, 36 m above the surface of a, lake, the angle of elevation of a bird is, , Observed to be 30° and angle of depression ot, 1s mage in the water of the lake is observed, , be 60. Find the actual, height of the bird, above the surface of the lake., , to, , 343, , B, , of, angles of depression, , as observed, , [2015], , two, , from the top of, , a, , ships, , A and, , light, , house, , If the, , 60° and 45 respectively., sides of the light, two ships are on the opposite, between the two ships., house, find the distance, whole, , 60, , m, , Gve, , high,, , your, , number., , are, , a n s w e r corect, , to the, , nearest, , 20171