Ella Walks 20 Feet Away From A 100-foot Tall Building. At What Angle Of Elevation Would She Have To Look Up To See The Top Of The Building? Round To The Nearest Tenth Of A Degree.

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Introduction

In this article, we will delve into the world of trigonometry and explore the concept of angles of elevation. We will use a real-world scenario to calculate the angle of elevation that Ella would need to look up to see the top of a 100-foot tall building from a distance of 20 feet away.

The Problem

Ella walks 20 feet away from a 100-foot tall building. At what angle of elevation would she have to look up to see the top of the building? This problem can be solved using the tangent function in trigonometry.

Understanding the Tangent Function

The tangent function is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. In this case, the opposite side is the height of the building (100 feet), and the adjacent side is the distance from the building (20 feet).

Calculating the Angle of Elevation

To calculate the angle of elevation, we can use the inverse tangent function (arctangent) to find the angle whose tangent is equal to the ratio of the opposite side to the adjacent side.

tan(θ) = opposite side / adjacent side tan(θ) = 100 / 20 tan(θ) = 5

Using a Calculator to Find the Angle

Using a calculator, we can find the angle whose tangent is equal to 5.

θ = arctan(5) θ ≈ 78.7°

Rounding to the Nearest Tenth of a Degree

To round the angle to the nearest tenth of a degree, we can use the following calculation:

θ ≈ 78.7° θ ≈ 78.7° (rounded to the nearest tenth of a degree)

Conclusion

In this article, we used the tangent function to calculate the angle of elevation that Ella would need to look up to see the top of a 100-foot tall building from a distance of 20 feet away. We found that the angle of elevation is approximately 78.7°, rounded to the nearest tenth of a degree.

Real-World Applications

The concept of angles of elevation has many real-world applications, including:

  • Surveying: Angles of elevation are used in surveying to determine the height of buildings, mountains, and other objects.
  • Architecture: Angles of elevation are used in architecture to design buildings and structures that are aesthetically pleasing and functional.
  • Aviation: Angles of elevation are used in aviation to determine the height of aircraft and to navigate through complex airspace.

Tips and Tricks

  • Use a calculator: When working with trigonometric functions, it's often easier to use a calculator to find the values of the functions.
  • Check your units: Make sure to check your units when working with trigonometric functions to ensure that you are using the correct units.
  • Use a diagram: Drawing a diagram can help you visualize the problem and make it easier to understand.

Common Mistakes

  • Not checking units: Failing to check units can lead to incorrect answers.
  • Not using a calculator: Not using a calculator can make it difficult to find the values of trigonometric functions.
  • Not drawing a diagram: Not drawing a diagram can make it difficult to visualize the problem and understand the solution.

Frequently Asked Questions

  • What is the angle of elevation? The angle of elevation is the angle between the horizontal and the line of sight to an object.
  • How do I calculate the angle of elevation? To calculate the angle of elevation, you can use the tangent function and the inverse tangent function (arctangent).
  • What are some real-world applications of angles of elevation? Angles of elevation have many real-world applications, including surveying, architecture, and aviation.
    Ella's Elevation Angle: A Mathematical Exploration =====================================================

Q&A: Angles of Elevation

Q: What is the angle of elevation?

A: The angle of elevation is the angle between the horizontal and the line of sight to an object.

Q: How do I calculate the angle of elevation?

A: To calculate the angle of elevation, you can use the tangent function and the inverse tangent function (arctangent). The formula is:

tan(θ) = opposite side / adjacent side

Q: What is the opposite side and the adjacent side in the formula?

A: In the formula, the opposite side is the height of the object, and the adjacent side is the distance from the object.

Q: How do I use a calculator to find the angle of elevation?

A: To use a calculator to find the angle of elevation, you can enter the value of the tangent function (tan(θ)) and press the "arctan" or "inverse tan" button to find the angle.

Q: What are some real-world applications of angles of elevation?

A: Angles of elevation have many real-world applications, including:

  • Surveying: Angles of elevation are used in surveying to determine the height of buildings, mountains, and other objects.
  • Architecture: Angles of elevation are used in architecture to design buildings and structures that are aesthetically pleasing and functional.
  • Aviation: Angles of elevation are used in aviation to determine the height of aircraft and to navigate through complex airspace.

Q: What are some common mistakes to avoid when calculating angles of elevation?

A: Some common mistakes to avoid when calculating angles of elevation include:

  • Not checking units: Failing to check units can lead to incorrect answers.
  • Not using a calculator: Not using a calculator can make it difficult to find the values of trigonometric functions.
  • Not drawing a diagram: Not drawing a diagram can make it difficult to visualize the problem and understand the solution.

Q: How do I draw a diagram to help me visualize the problem?

A: To draw a diagram, you can draw a right triangle with the height of the object as the opposite side and the distance from the object as the adjacent side. Then, you can draw a line from the object to the point where the line of sight intersects the ground.

Q: What are some tips and tricks for working with angles of elevation?

A: Some tips and tricks for working with angles of elevation include:

  • Use a calculator: When working with trigonometric functions, it's often easier to use a calculator to find the values of the functions.
  • Check your units: Make sure to check your units when working with trigonometric functions to ensure that you are using the correct units.
  • Use a diagram: Drawing a diagram can help you visualize the problem and make it easier to understand.

Q: What are some common misconceptions about angles of elevation?

A: Some common misconceptions about angles of elevation include:

  • Thinking that the angle of elevation is always 90 degrees: The angle of elevation is not always 90 degrees. It depends on the height of the object and the distance from the object.
  • Thinking that the angle of elevation is always the same: The angle of elevation can vary depending on the height of the object and the distance from the object.

Q: How do I apply angles of elevation in real-world situations?

A: To apply angles of elevation in real-world situations, you can use the following steps:

  1. Identify the height of the object: Determine the height of the object that you want to measure.
  2. Identify the distance from the object: Determine the distance from the object that you want to measure.
  3. Use the tangent function: Use the tangent function to calculate the angle of elevation.
  4. Use a calculator: Use a calculator to find the value of the tangent function.
  5. Draw a diagram: Draw a diagram to help you visualize the problem and understand the solution.

Conclusion

In this article, we have explored the concept of angles of elevation and how to calculate them using the tangent function and the inverse tangent function (arctangent). We have also discussed some real-world applications of angles of elevation and some tips and tricks for working with them.