Example Problem Of Angle Of Elevation

Angle of Elevation Problems: Real-World Applications and Solutions

Angle of elevation problems are a staple in trigonometry, bridging the gap between theoretical concepts and practical applications. Understanding these problems is crucial for fields ranging from surveying and construction to navigation and astronomy. This comprehensive guide delves into various examples of angle of elevation problems, exploring different solution strategies and highlighting the importance of accurate measurements and calculations.

Understanding Angle of Elevation

Before we dive into specific problems, let's establish a clear understanding of the term "angle of elevation." The angle of elevation is the angle formed between the horizontal line of sight and the line of sight upward to an object. Imagine you're standing on the ground looking up at a bird; the angle formed between the horizontal line of your gaze and your line of sight to the bird is the angle of elevation. It's always measured from the horizontal upwards.

Key Concepts for Solving Angle of Elevation Problems:

• Right-angled triangles: Most angle of elevation problems involve right-angled triangles, leveraging trigonometric functions (sine, cosine, tangent) to find unknown sides or angles.
• Trigonometric ratios: Understanding the relationships between the sides and angles of a right-angled triangle (SOH CAH TOA) is fundamental.
• Units: Ensure consistent use of units (degrees for angles, meters or feet for lengths) throughout the problem.
• Accurate measurements: The accuracy of the solution depends heavily on the accuracy of the initial measurements.

Example Problems: A Gradual Progression

Let's explore a range of angle of elevation problems, starting with simpler scenarios and progressing to more complex ones.

Problem 1: The Simple Building

A building casts a shadow 20 meters long. The angle of elevation from the tip of the shadow to the top of the building is 60 degrees. What is the height of the building?

Solution:

This problem forms a right-angled triangle. The shadow is the adjacent side, the building's height is the opposite side, and the angle of elevation is 60 degrees. We use the tangent function:

tan(60°) = opposite / adjacent

tan(60°) = height / 20 meters

height = 20 meters * tan(60°)

height ≈ 34.64 meters

Therefore, the building is approximately 34.64 meters tall.

Problem 2: The Two Buildings

Two buildings are 50 meters apart. From the roof of the shorter building, the angle of elevation to the top of the taller building is 30 degrees, and the angle of depression to the base of the taller building is 45 degrees. Find the height of each building.

Solution:

This problem involves two right-angled triangles. Let's denote the height of the shorter building as 'h1' and the height of the taller building as 'h2'.

Triangle 1 (Angle of Elevation):

tan(30°) = (h2 - h1) / 50 meters

h2 - h1 = 50 meters * tan(30°)

h2 - h1 ≈ 28.87 meters

Triangle 2 (Angle of Depression):

tan(45°) = h1 / 50 meters

h1 = 50 meters * tan(45°)

h1 = 50 meters

Substituting h1 into the first equation:

h2 - 50 meters ≈ 28.87 meters

h2 ≈ 78.87 meters

Therefore, the shorter building is approximately 50 meters tall, and the taller building is approximately 78.87 meters tall.

Problem 3: The Airplane's Altitude

An airplane is flying at a constant altitude. From a point on the ground 1000 meters from the airplane's projection on the ground, the angle of elevation to the airplane is 15 degrees. What is the airplane's altitude?

Solution:

This is a straightforward right-angled triangle problem. The horizontal distance is the adjacent side, the altitude is the opposite side, and the angle of elevation is 15 degrees. Using the tangent function:

tan(15°) = altitude / 1000 meters

altitude = 1000 meters * tan(15°)

altitude ≈ 267.95 meters

The airplane's altitude is approximately 267.95 meters.

Problem 4: The Inclined Plane and the Flagpole

A flagpole stands on an inclined plane that makes an angle of 10 degrees with the horizontal. From a point 20 meters down the incline, the angle of elevation to the top of the flagpole is 30 degrees. What is the height of the flagpole?

Solution:

This problem requires a more nuanced approach. We can break it down into two right-angled triangles.

Triangle 1 (Inclined Plane):

The distance down the incline (20 meters) is the hypotenuse. The horizontal distance is the adjacent side.

cos(10°) = adjacent / 20 meters

adjacent ≈ 19.70 meters

Triangle 2 (Flagpole):

The horizontal distance (from Triangle 1) is the adjacent side. The height of the flagpole is the opposite side, and the angle of elevation is 30 degrees.

tan(30°) = height / 19.70 meters

height = 19.70 meters * tan(30°)

height ≈ 11.38 meters

Therefore, the flagpole is approximately 11.38 meters tall.

Problem 5: The Lighthouse and the Two Ships

From the top of a lighthouse 50 meters above sea level, the angles of depression to two ships are 20 degrees and 10 degrees. If the ships are on a line with the lighthouse, how far apart are the ships?

Solution:

This problem involves two right-angled triangles sharing a common side (the lighthouse height).

Triangle 1 (Closer Ship):

tan(20°) = 50 meters / distance1

distance1 = 50 meters / tan(20°)

distance1 ≈ 137.37 meters

Triangle 2 (Further Ship):

tan(10°) = 50 meters / distance2

distance2 = 50 meters / tan(10°)

distance2 ≈ 283.56 meters

The distance between the ships is the difference between distance2 and distance1:

distance_between_ships = distance2 - distance1 ≈ 283.56 meters - 137.37 meters ≈ 146.19 meters

The ships are approximately 146.19 meters apart.

Advanced Considerations and Applications

While the problems above demonstrate basic applications of angle of elevation, real-world scenarios often involve more complex geometries and additional factors.

Multiple Angles of Elevation

Problems might involve multiple angles of elevation from different points, requiring the simultaneous solution of multiple trigonometric equations.

Obstructions and Irregular Terrain

Real-world measurements may be affected by obstructions or irregular terrain, necessitating adjustments to calculations and potentially employing more advanced surveying techniques.

Applications in Different Fields

• Surveying: Determining heights of inaccessible objects like mountains or trees.
• Construction: Calculating the height and angles of structures during design and construction.
• Navigation: Determining distances and altitudes in air and sea navigation.
• Astronomy: Calculating distances to celestial objects.
• Military: Range finding and trajectory calculations.

Mastering Angle of Elevation Problems

Consistent practice and a strong understanding of trigonometric principles are crucial for mastering angle of elevation problems. Start with simpler problems like the ones outlined above, gradually increasing complexity as your understanding grows. Remember to always:

• Draw a diagram: Visualizing the problem using a clear diagram simplifies the solution process.
• Identify the known and unknown quantities: This helps in selecting the appropriate trigonometric function.
• Double-check your calculations: Ensuring accuracy is paramount, especially in applications where precision is critical.

By practicing these strategies and deepening your understanding of the concepts involved, you'll become proficient in solving a wide variety of angle of elevation problems and appreciating their significant role in numerous real-world applications.