The Observer Moves Towards the Tower

Oct 13, 2025 by miodeya.com 104 views

Introduction

The angle of elevation is a fundamental concept in trigonometry, which is used to calculate the height of objects or the distance between them. In this article, we will explore the concept of the angle of elevation and use it to find the height of a tower. We will use the given information about the angle of elevation and the distance moved by the observer to calculate the height of the tower.

The Angle of Elevation

The angle of elevation is the angle between the horizontal and the line of sight to the top of an object. It is measured in degrees and is used to calculate the height of the object. In this case, we are given that the angle of elevation of the top of a tower from a point on the ground is $$30^{\circ}$$.

The Observer Moves Towards the Tower

When the observer moves 20 m towards the tower, the angle of elevation becomes $$45^{\circ}$$. This means that the distance between the observer and the tower has decreased by 20 m.

Using Trigonometry to Find the Height of the Tower

We can use the concept of trigonometry to find the height of the tower. Let's denote the height of the tower as $$h$$ and the distance between the observer and the tower as $$d$$. We can use the tangent function to relate the angle of elevation, the height of the tower, and the distance between the observer and the tower.

\tan(30^{\circ}) = \frac{h}{d}

\tan(45^{\circ}) = \frac{h}{d-20}

Solving the Equations

We can solve the two equations to find the value of $$h$$.

\tan(30^{\circ}) = \frac{h}{d}

\tan(45^{\circ}) = \frac{h}{d-20}

We can multiply both sides of the first equation by $$d$$ and both sides of the second equation by $$d-20$$ to get:

h = d\tan(30^{\circ})

h = (d-20)\tan(45^{\circ})

Equating the Two Expressions for $$h$$

We can equate the two expressions for $$h$$ to get:

d\tan(30^{\circ}) = (d-20)\tan(45^{\circ})

Solving for $$d$$

We can solve for $$d$$ by expanding the equation and simplifying:

d\tan(30^{\circ}) = d\tan(45^{\circ}) - 20\tan(45^{\circ})

d\tan(30^{\circ}) - d\tan(45^{\circ}) = -20\tan(45^{\circ})

d(\tan(30^{\circ}) - \tan(45^{\circ})) = -20\tan(45^{\circ})

d = \frac{-20\tan(45^{\circ})}{\tan(30^{\circ}) - \tan(45^{\circ})}

Calculating the Value of $$d$$

We can calculate the value of $$d$$ by plugging in the values of the trigonometric functions:

d = \frac{-20\tan(45^{\circ})}{\tan(30^{\circ}) - \tan(45^{\circ})}

d = \frac{-20(1)}{0.57735 - 1}

d = \frac{-20}{-0.42265}

d = 47.24

Calculating the Value of $$h$$

We can calculate the value of $$h$$ by plugging in the value of $$d$$ into one of the original equations:

h = d\tan(30^{\circ})

h = 47.24 \times 0.57735

h = 27.27

Conclusion

In this article, we used the concept of the angle of elevation to find the height of a tower. We were given that the angle of elevation of the top of a tower from a point on the ground is $$30^{\circ}$$ and that when the observer moves 20 m towards the tower, the angle of elevation becomes $$45^{\circ}$$. We used the tangent function to relate the angle of elevation, the height of the tower, and the distance between the observer and the tower. We solved the equations to find the value of $$h$$, which is the height of the tower.

References

[1] "Trigonometry" by Michael Corral
[2] "Mathematics for Engineers and Scientists" by Donald R. Hill

Introduction

In our previous article, we explored the concept of the angle of elevation and used it to find the height of a tower. In this article, we will answer some of the most frequently asked questions about the angle of elevation and its applications.

Q: What is the angle of elevation?

A: The angle of elevation is the angle between the horizontal and the line of sight to the top of an object. It is measured in degrees and is used to calculate the height of the object.

Q: How is the angle of elevation used in real-life situations?

A: The angle of elevation is used in a variety of real-life situations, including:

Calculating the height of buildings and towers
Determining the distance between objects
Measuring the angle of a slope or incline
Calculating the height of mountains and hills

Q: What are some common applications of the angle of elevation?

A: Some common applications of the angle of elevation include:

Surveying and mapping
Architecture and engineering
Physics and astronomy
Navigation and aviation

Q: How do I calculate the angle of elevation?

A: To calculate the angle of elevation, you can use the following formula:

\tan(\theta) = \frac{h}{d}

where $$\theta$$ is the angle of elevation, $$h$$ is the height of the object, and $$d$$ is the distance between the observer and the object.

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

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

Not accounting for the height of the observer
Not using the correct formula
Not considering the effects of refraction and reflection
Not using a calculator or trigonometric table

Q: Can I use the angle of elevation to calculate the distance between objects?

A: Yes, you can use the angle of elevation to calculate the distance between objects. However, you will need to know the height of one of the objects and the angle of elevation.

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

A: To use the angle of elevation to calculate the height of an object, you can use the following formula:

h = d\tan(\theta)

where $$h$$ is the height of the object, $$d$$ is the distance between the observer and the object, and $$\theta$$ is the angle of elevation.

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

A: Some real-world examples of the angle of elevation include:

Calculating the height of a skyscraper
Determining the distance between two mountains
Measuring the angle of a slope or incline
Calculating the height of a satellite in orbit around the Earth

Q: Can I use the angle of elevation to calculate the angle of a slope or incline?

A: Yes, you can use the angle of elevation to calculate the angle of a slope or incline. However, you will need to know the height of the slope or incline and the distance between the observer and the slope or incline.

Q: How do I use the angle of elevation to calculate the angle of a slope or incline?

A: To use the angle of elevation to calculate the angle of a slope or incline, you can use the following formula:

\theta = \arctan\left(\frac{h}{d}\right)

where $$\theta$$ is the angle of the slope or incline, $$h$$ is the height of the slope or incline, and $$d$$ is the distance between the observer and the slope or incline.

Conclusion

In this article, we have answered some of the most frequently asked questions about the angle of elevation and its applications. We have also provided some real-world examples of the angle of elevation and explained how to use it to calculate the height of objects, the distance between objects, and the angle of slopes or inclines.

References

[1] "Trigonometry" by Michael Corral
[2] "Mathematics for Engineers and Scientists" by Donald R. Hill
[3] "The Angle of Elevation" by Math Open Reference
[4] "Trigonometry" by Khan Academy