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Chapter Analysis
Intermediate11 pages • EnglishQuick Summary
This chapter focuses on the practical aspects of trigonometry related to heights and distances. It introduces the concepts of angle of elevation and angle of depression, emphasizing their application in real-world scenarios like determining the height of towers, buildings, and distances across rivers. The chapter illustrates the use of trigonometric ratios to solve such problems, and examples help in understanding the application of these concepts.
Key Topics
- •Angle of Elevation
- •Angle of Depression
- •Trigonometric Ratios
- •Solving Real-world Problems
- •Heights and Distances
- •Using Trigonometric Tables
- •Applications in Daily Life
- •Understanding Right Triangles
Learning Objectives
- ✓Understand and apply the concept of angle of elevation and depression.
- ✓Use trigonometric ratios to solve height and distance problems.
- ✓Develop problem-solving skills using real-world scenarios.
- ✓Apply trigonometry to determine heights of objects and distances.
- ✓Interpret and construct diagrams for solving trigonometric problems.
- ✓Use appropriate trigonometric tables where necessary.
Questions in Chapter
A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is 30°.
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A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30° with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.
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A contractor plans to install two slides for the children to play in a park. For the children below the age of 5 years, she prefers to have a slide whose top is at a height of 1.5 m, and is inclined at an angle of 30° to the ground, whereas for elder children, she wants to have a steep slide at a height of 3m, and inclined at an angle of 60° to the ground. What should be the length of the slide in each case?
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The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower, is 30°. Find the height of the tower.
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A kite is flying at a height of 60 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is 60°. Find the length of the string, assuming that there is no slack in the string.
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A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eyes to the top of the building increases from 30° to 60° as he walks towards the building. Find the distance he walked towards the building.
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From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a 20 m high building are 45° and 60° respectively. Find the height of the tower.
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A statue, 1.6 m tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60° and from the same point the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal.
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The angle of elevation of the top of a building from the foot of the tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 50 m high, find the height of the building.
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Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60° and 30°, respectively. Find the height of the poles and the distances of the point from the poles.
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A TV tower stands vertically on a bank of a canal. From a point on the other bank directly opposite the tower, the angle of elevation of the top of the tower is 60°. From another point 20 m away from this point on the line joining this point to the foot of the tower, the angle of elevation of the top of the tower is 30°. Find the height of the tower and the width of the canal.
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From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 45°. Determine the height of the tower.
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As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.
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A 1.2 m tall girl spots a balloon moving with the wind in a horizontal line at a height of 88.2 m from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is 60°. After some time, the angle of elevation reduces to 30°. Find the distance travelled by the balloon during the interval.
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A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30°, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be 60°. Find the time taken by the car to reach the foot of the tower from this point.
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Additional Practice Questions
A flagpole casts a shadow 10 m long when the angle of elevation of the Sun is 45°. What is the height of the flagpole?
easyAnswer: Since the angle of elevation of the sun is 45°, the height of the flagpole is equal to the length of the shadow. Therefore, the height of the flagpole is 10 meters.
A boy is flying a kite at a height of 50 m, with the string making an angle of 60° with the ground. Find the length of the string assuming it is taut.
mediumAnswer: Using trigonometry, length of the string = height / sin(60°) = 50 / (√3/2) = 100/√3 ≈ 57.74 meters.
A ladder leaning against a wall makes an angle of 30° with the ground. If the ladder reaches a height of 4 m on the wall, how long is the ladder?
mediumAnswer: By sine rule, sin(30°) = opposite/hypotenuse = 4/ladder length. Thus, ladder length = 4/sin(30°) = 8 meters.
Two buildings are 50 m apart. From the top of the first building, the angles of elevation and depression to the top and bottom of the second building are 30° and 60°, respectively. Find the height of the second building.
hardAnswer: Height of the second building = distance * (tan(30°) + tan(60°)) = 50 * (1/√3 + √3) ≈ 50 * 1.865 = 93.25 meters.
A person standing 40 m away from a building observes the top of the building at an angle of elevation of 50°. Find the height of the building.
mediumAnswer: Height of the building = distance * tan(50°) = 40 * 1.1918 ≈ 47.67 meters.