Mathematics, grade -10 297 Example 1 A man of 1.80 meter high observes the angle of elevation at the top of tree and found it to be 45°. If the distance of the man and the tree is 20 meter, find the height of the tree. Solution When the man observes to the top of the tree, the angle between the line of sight and the line parallel to the ground is 45°. In the figure, the height of the man is 1.80 meter and the total height of the tree = (x + 1.80) meter In the right angled triangle ABC We have, tan45º = x 20 or, 1= x 20 or, x = 20 m Hence the height of tree = 20 + 1.80 = 21.80 meter. of the tree from us i.e. (ED), then we can find the height of the tree. Let's look at the top of the tree with a Clinometer. The angle of elevation of the top of the tree is found to be 45°. If the height of your eyes (CE) = 1.6 m, the angle of elevation ∠AED = 45° and the distance from you to the bottom of the tree is BC = DE = 30 m then, From the right angled triangle ADE tan45º = AD 30 or, 1= AD 30 or, AD = 30 m Hence, the total height of the tree = 30 + 1.60 = 31.60 m. A B C x m 20 m 1.80 m 1.80 m 45º
298 Mathematics, grade -10 Example 2 A man finds the angle of elevation of the top of a tower to be 60°. The height of tower is 140 m and the distance between man and tower is x m. Find the value of x. Solution Here, Height of tower (AC) = 140 m The distance between man and tower (BC) = x m The angle of elevation ∠ABC = 60° In right angled triangle ACB, We have, tan60° = AC BC or, 3 = 140 x or, x 3 = 140 or, x = 140 1.732 ⸫ BC = 80.83 m Hence, the distance between man and tower is 80.83 60º C B A x m 140 m Example 3 A tree 18 m high is broken by the wind so that its top touches the ground and makes an angle of 30° with the ground. Find the length of broken part of the tree. Solution Let, the height of tree (AB) = 18 m The length of broken part of tree (AD) = CD = x m The height of remaining part of tree (BD) = (18 - x) m The angle making by broken part of tree = ∠DCB = 30° Now, In right angled triangle CBD, We have, sin30° = BD CD or, 1 2 = 18 – x x or, x = 36 – 2x or, x + 2x = 36 or, 3x = 36 or, x = 12 m Hence, the length of broken part of the tree is 12 m A 30º D B C (18 – x m) 18 m x m x m
Mathematics, grade -10 299 Example 4 The distance between a tower and a house is one third of the height of the tower. If the height of the tower is 60 m and the angle of depression from the top of the tower to the house is 45°, find the height of the house. Solution Let, the height of the tower (AB) = 60 m DE be the height of the house BE be the distance between the tower and house. The distance between the tower and house; BE = CD = 60 ×1 3 = 20 m We know that, DE = BC and by alternate angle = ∠FAD = 45° Now, in the right angled triangle ACD, We have, tan45° = AC CD or, 1 = AC 20 or, AC = 20 m Again, BC = DE = AB - AC = 60 - 20 = 40 m Hence, the height of the is 40 m. Example 5 A man 1.2 m tall observes the angle of elevation at the top of a tower and finds to be 60°. If the height of the tower is 53.2 m, find the distance between the tower and the man. Solution Let, the height of the tower (AB) = 53.2 m The height of the man (DE) = 1.2 m BE be the distance between the tower and the man The angle of elevation ∠ADC = 60° We know that, DE = BC and BE = CD (\ Being opposite sides of a rectangle) Now, in the right angled triangle ACD, We have, tan60º = AC CD . 60º 53.2 m52 m B E C D A 1.2 m 60 m A C B E D 20 m 45º 45º F
300 Mathematics, grade -10 or, 3 = AB – BC CD or, 3 = 53.2 – 1.2 CD or, 3 = 52 CD or, 3 × CD = 52 or, CD = 52 3 or, CD = 52 1.732 or, CD = 30.02 m Hence, the distance between the tower and the man is 30. 02 m. Example 6 The diameter of a circular pond is 100 m. A pole is fixed at the centre of the pond and the height of the pole above the water surface is 50 m, what is the angle of elevation to the top of the pole observed from the edge of the pond? Find it. Solution Let, the diameter of the pond (BD) = d = 100 m The radius of the pond (OB) = d/2 = 100/2 = 50 m AO = 50 m be the height of the pole above the water surface. The angle of elevation from B to A ∠ABO = θ Now, in right angled triangle AOB, We have, tanθ = OA OB = 50 50 = 1 or, tanθ = tan45º ⸫ θ = 45º Hence, the angle of elevation to the top of pole observed from the edge of the pond is 45°. 100 m 50m 0 D B A θ
Mathematics, grade -10 301 Exercise 15 1. (a) Define the angle of elevation and depression with example. (b) In a right angled triangle, which trigonometric ratio is related to both perpendicular and base? (c) In a right angled triangle, which trigonometric ratio has the relation with perpendicular and hypotenuse? 2. Find the value of x on the given right-angled triangles. (a) 60º B C A x m 5 3 (b) x 30º B C A 5 3 (c) xº B C A 10 m 10 m 3. (a) The height of a tower is 60 m and the distance between a man and a tower is x m. A man finds the angle of elevation of the top of a tower to be 30°. Find the value of x. (b) The height of tower is 12 m and the distance between man and tower is 12 m. The man finds the angle at elevation of the top of the tower to be x°. Find the value of x. (c) The height of a tower is x m and the distance between a man and the tower is 12 m. The man finds the angle of elevation at the top of the tower to be 45°. Find the value of x. 4. (a) A tree of 14 m high is broken by the wind so that its top touches the ground (not separated from the main stem and makes an angle of 60° with the ground. Find the length of the broken part of the tree. (b) A tree is broken from the middle part by the wind so that it's top touches the ground (not separated from the main stem and makes an angle of 60° with the ground. If the length of the broken part of the tree is 7.5 m, find the height of the tree before it was broken. (c) A tree is broken at the middle part by the wind so that it's top touches the ground (not separated from the main stem and makes an angle of 30° with the ground. The length of the broken part of the tree is 30 m.
302 Mathematics, grade -10 (i) Find the height of the tree before it was broken. ii) How far does it meet the ground level from the base of the tree? 5. (a) A man of 1.7 m height observes the angle of elevation at the top of a tower and finds it to be 60°. If the distance between the tower and the man is 30 m, find the height of the tower. (b) A man of height 2 m is flying a kite from the roof of a house of 33.6 m. If the length of the string of the kite is 90√2 m and it makes an angle of 45°with the horizon, find the height of kite from the ground level. (c) A 1.5 m tall man observes the angle of elevation at the top of a tree of 51.5 m height and finds it to be 45°, find the distance between the tree and the man. 6. (a) The distance between a tower and a man is 20 m. The height of the ree is 36.5 m. If the angle of elevation from the eye of the man to the top of the tower is 60°, find the height of the man. (b) From the top of a house of 30 ft hight, the angle of depression at the top of a tree is 30°. If the distance between the house and the tree is 10√3 m, find the height of the tree. (c) The height of tower is 60 m and the distance between the tower and house is 35 m. If a man observes the angle of depression from the top of tower to the top of house is 45°, find the height of house. 7. (a) The diameter of a circular pond is 90 m. A pole is fixed at the centre of the pond and height of the pole above the water surface is 45 m, what is the angle of elevation of the top of pole observed from the edge of the pond? Find it. (b) The diameter of a circular pond is 130 m. A pole is fixed at the centre of the pond. A person finds the angle of elevation of the top of the pole observed from the edge of the pond is 45°. What is the height of the pole above the water surface? Find it. (c) At the centre of a circular pond, there is a pole of 11.62 m height above the surface of water. From a point on the edge of the pond, a man of 1.62 m height observed the angle of elevation at of the top of the pole and found it to be 30°. Find the diameter of the pond. 8. (a) On the occasion of a festival, Ramesh is flying a kite. The thread of the kite makes an angle of 30° with the horizon. If the length of the thread is 120 m and the Ramesh's height is 1.5 m, find the height of the kite from the ground.
Mathematics, grade -10 303 (b) 1.5 m tall Ramsharan is flying a kite from the roof of a house of 9 m height. The string of the kite makes an angle of 30° with the horizon. If the height of the kite from the ground is 58 m, what is the length of the string. Find it. (c) A man of height 2 m is flying a kite from the roof of a house of 32 m high. If the length of the string of the kite is 66√2 m and makes an angle of 60° with the horizon, work out to calculate the height of the kite from the ground? 9. The length of the shadow of a pole of 20 m high at 2 pm is 20√3 m. on the meantime. Find the length of the shadow of the tower with the height 25√3 m? 10. A tree 25 m high is just in the one corner of the ground of a school. A man of height 1.2 m is sitting on another corner of the ground. The distance between the man and the tree is 23.8 m. (a) What is the angle of elevation at the top of the tree made by the man? (b) Does the angle of elevation increase when the height of the tree is increases? Write with reason. (c) Is the angle of elevation whether increases or decreases when the distance between the man and tree is decreases. Write with reason. 11. In the given figure two buildings 20 m and 32 m are shown. The distance between then is 12 m. (a) What is the name of the the angle formed with the horizontal line when the parrot looks at the cat? (b) Find the angle in degree with the horizontal line when the parrot looks at the cat? (c) What is the distance between the pole and cat? (d) Does the angle the parrot makes when looking at the cat decrease as the cat moves towards the pole? Write with reason. 12. The circumference of a circular pond is 176 m. A pole is fixed at the centre of the pond. From a point on the edge of the pond, a man of 1.6 m tall observed the angle of elevation of the top of the pole and found it be 45°. 2 3 m
304 Mathematics, grade -10 (a) Find the distance between the man and the pole. (b) What is the height of the pole above the water surface? Find it. (c) How much less should the height of the pole above the water surface such that it would have made the angle of elevation of 30°? 13. A man of height 1.2 m is flying a kite form the roof of a house 8.8 m high. If the length of the string of the kite is 180 m and makes an angle of 30° with the horizon, (a) Represent the given relationship in a diagrammatical form. (b) What is the height of the kite in meter from the ground? Find it. (c) What is thecheck distance between the man and the kite? Find it. 14. In the figure, two buildings 20 m and 32 m high are shown. The distance between them is 12 m. (a) What type of angle is formed when observed from the roof of tall building to the roof of the short building? (b) Explain with reason the relation between the angle that forms when you observes the top of the tall building from the top of the short building and the top of the short building from the top of the big building. (c) Calculate the angle in degree, when an observer observes from the roof of the small building to the roof of big building. (d) If the ladder is put from the top of the small building to the top of the big building, what should be the length of the ladder? Calculate it. 15. The tower and the house are on the same ground level and distance between them is 25 m. The height of the house is 15 m. (a) The angle of elevation from a point A of the roof of the house to the top of tower B is 45°, what is the height of the tower? (b) From the figure, how the angle of elevation changes if the height of the tower increases. The angle of depression of a house 20 m to the east of the tower is 60° when observed from top of the tower of height 50 m.
Mathematics, grade -10 305 Answer 2. (a) 5 m (b) 5 m (c) 45º 3. (a) 60 3 m (b) 45º (c) 12 m 4. (a) 7.5 m (b) 14 m (c) 45 m, 15 3 m 5. (a) 53.6 m (b) 125 m (c) 50 m 6. (a) 1.86 m (b) 20 ft (c) 25 m 7. (a) 45º (b) 65 m (c) 11.55 m 8. (a) 61.5 m (b) 95 m (c) 100 m 9. 30º, 75 m 10. (a) 45º 11. (a) 30º (b) 6 m 12. (a) 28 m (b) 29 m (c) 11.83 m 13. (a) 100 m (c) 90 3 m 14. (a) and (b) Show ot to your teacher (a) 45º (d) 12 2 m 15. (a) 40 m 16. (a) 60º (b) 34.64 m (c) 15.36 m (d) 14.64 m Project work Make a group of friends and find out the things in the highest and lowest areas at around your house. Find out practically the angles with respect to their height and distance or the height with respect to their angles and distance. Discuss the result obtained thus and find out the conclusion, and present it in the class. 16. From the top of the 50 m hight tower an angle of 60° is formed when looking at the roof of a house of 20 m to the east of the tower. (a) What is the angle formed when observed from a point A on the roof of the house to the top of the view tower? (b) What is the height of the part BC of the tower? Find it. (c) What is the height of the house? Find it. (d) What distance does she have to come down from the top of the tower to observe the terrace (top) of the house such that the angle of depression of 45°
306 Mathematics, grade -10