A Person Is Standing Exactly 36 Ft From A Telephone Pole. There Is A $30^{\circ}$ Angle Of Elevation From The Ground To The Top Of The Pole.What Is The Height Of The Pole?A. 12 FtB. $12 \sqrt{3}$ FtC. 18 FtD. $18

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Introduction

In trigonometry, the angle of elevation is a fundamental concept used to calculate the height of objects or structures. In this article, we will explore how to use the angle of elevation to determine the height of a telephone pole. We will use a real-world scenario to demonstrate the application of trigonometric principles.

The Problem

A person is standing exactly 36 ft from a telephone pole. There is a $30^{\circ}$ angle of elevation from the ground to the top of the pole. What is the height of the pole?

Understanding the Angle of Elevation

The angle of elevation is the angle between the ground and the line of sight to the top of the object. In this case, the angle of elevation is $30^{\circ}$. This angle is measured from the ground, not from the horizontal line connecting the person to the base of the pole.

Using Trigonometry to Calculate the Height

To calculate the height of the pole, we can use the tangent function, which is defined as the ratio of the opposite side (the height of the pole) to the adjacent side (the distance from the person to the base of the pole).

tan⁡(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}

In this case, the angle $\theta$ is $30^{\circ}$, the opposite side is the height of the pole (h), and the adjacent side is the distance from the person to the base of the pole (36 ft).

tan⁡(30∘)=h36\tan(30^{\circ}) = \frac{h}{36}

Solving for the Height

To solve for the height (h), we can multiply both sides of the equation by 36.

h=36×tan⁡(30∘)h = 36 \times \tan(30^{\circ})

Using a calculator, we can find that $\tan(30^{\circ}) = \frac{1}{\sqrt{3}}$.

h=36×13h = 36 \times \frac{1}{\sqrt{3}}

To simplify the expression, we can multiply the numerator and denominator by $\sqrt{3}$.

h=3633h = \frac{36\sqrt{3}}{3}

h=123h = 12\sqrt{3}

Conclusion

In this article, we used the angle of elevation to calculate the height of a telephone pole. We applied the tangent function to solve for the height, and we found that the height of the pole is $12\sqrt{3}$ ft.

Answer

The correct answer is:

  • B. $12 \sqrt{3}$ ft

Discussion

This problem is a classic example of how trigonometry can be used to solve real-world problems. The angle of elevation is a fundamental concept in trigonometry, and it is used in a wide range of applications, from architecture to engineering.

In this problem, we used the tangent function to calculate the height of the pole. However, we could have also used the sine or cosine function to solve the problem. The choice of function depends on the specific problem and the information given.

Additional Examples

Here are a few additional examples of how the angle of elevation can be used to calculate the height of objects or structures:

  • A person is standing 25 ft from a building. There is a $45^{\circ}$ angle of elevation from the ground to the top of the building. What is the height of the building?
  • A person is standing 15 ft from a tree. There is a $60^{\circ}$ angle of elevation from the ground to the top of the tree. What is the height of the tree?
  • A person is standing 30 ft from a mountain. There is a $30^{\circ}$ angle of elevation from the ground to the top of the mountain. What is the height of the mountain?

Q&A: Calculating the Height of Objects Using the Angle of Elevation

Introduction

In our previous article, we explored how to use the angle of elevation to calculate the height of a telephone pole. In this article, we will provide a Q&A section to help you better understand the concept and how to apply it to real-world problems.

Q: What is the angle of elevation?

A: The angle of elevation is the angle between the ground and the line of sight to the top of the object. It is measured from the ground, not from the horizontal line connecting the person to the base of the object.

Q: How do I calculate the height of an object using the angle of elevation?

A: To calculate the height of an object, you can use the tangent function, which is defined as the ratio of the opposite side (the height of the object) to the adjacent side (the distance from the person to the base of the object).

Q: What is the formula for calculating the height of an object using the angle of elevation?

A: The formula is:

h=distance×tan⁡(θ)h = \text{distance} \times \tan(\theta)

where h is the height of the object, distance is the distance from the person to the base of the object, and θ is the angle of elevation.

Q: How do I find the tangent of an angle?

A: You can find the tangent of an angle using a calculator or by using a trigonometric table. Alternatively, you can use the following formula:

tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}

Q: What is the difference between the angle of elevation and the angle of depression?

A: The angle of elevation is the angle between the ground and the line of sight to the top of the object, while the angle of depression is the angle between the ground and the line of sight to the bottom of the object.

Q: How do I calculate the height of an object using the angle of depression?

A: To calculate the height of an object using the angle of depression, you can use the same formula as before, but with the angle of depression instead of the angle of elevation.

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

A: The angle of elevation has many real-world applications, including:

  • Architecture: to calculate the height of buildings and other structures
  • Engineering: to calculate the height of bridges and other infrastructure
  • Surveying: to calculate the height of landmarks and other features
  • Photography: to calculate the height of objects in a photograph

Q: What are some common mistakes to avoid when calculating the height of an object using the angle of elevation?

A: Some common mistakes to avoid include:

  • Measuring the angle of elevation incorrectly
  • Measuring the distance incorrectly
  • Using the wrong formula or trigonometric function
  • Not accounting for the height of the observer

Conclusion

In this article, we provided a Q&A section to help you better understand the concept of the angle of elevation and how to apply it to real-world problems. We hope this article has been helpful in clarifying any questions you may have had about the angle of elevation.

Additional Resources

For more information on the angle of elevation, we recommend the following resources:

  • Trigonometry textbooks and online resources
  • Online calculators and trigonometric tables
  • Real-world examples and case studies

Practice Problems

To practice calculating the height of objects using the angle of elevation, try the following problems:

  • A person is standing 25 ft from a building. There is a $45^{\circ}$ angle of elevation from the ground to the top of the building. What is the height of the building?
  • A person is standing 15 ft from a tree. There is a $60^{\circ}$ angle of elevation from the ground to the top of the tree. What is the height of the tree?
  • A person is standing 30 ft from a mountain. There is a $30^{\circ}$ angle of elevation from the ground to the top of the mountain. What is the height of the mountain?

We hope these practice problems help you to better understand the concept of the angle of elevation and how to apply it to real-world problems.