Calculating the Height of the Mountain

Oct 13, 2025 by miodeya.com 141 views

Introduction

In trigonometry, the angle of elevation and depression are used to calculate the height of objects or mountains. The angle of elevation is the angle between the horizontal and the line of sight to the top of the object, while the angle of depression is the angle between the horizontal and the line of sight to the base of the object. In this article, we will use the angles of elevation and depression to calculate the height of a mountain.

Given Information

The angle of elevation from the top of a mast to the top of the mountain is 30°.
The angle of depression from the top of a mast to the base of the mountain is 60°.
The height of the mast is 40m.

Calculating the Height of the Mountain

To calculate the height of the mountain, we need to use the concept of trigonometry. We can use the tangent function to relate the angle of elevation, the height of the mast, and the height of the mountain.

Let's denote the height of the mountain as h. We can draw a right-angled triangle with the mast as the adjacent side, the line of sight to the top of the mountain as the opposite side, and the line of sight to the base of the mountain as the adjacent side.

Using the tangent function, we can write:

tan(30°) = (h - 40) / x

where x is the distance from the mast to the base of the mountain.

We can also write:

tan(60°) = 40 / x

Solving for x, we get:

x = 40 / tan(60°)

Now, we can substitute the value of x into the first equation:

tan(30°) = (h - 40) / (40 / tan(60°))

Simplifying the equation, we get:

tan(30°) = (h - 40) / (40 / √3)

Multiplying both sides by (40 / √3), we get:

(h - 40) = 40 * tan(30°) / √3

Simplifying the equation, we get:

h - 40 = 40 * 1/√3 / √3

h - 40 = 40 * 1/3

h - 40 = 40/3

Adding 40 to both sides, we get:

h = 40 + 40/3

h = (120 + 40) / 3

h = 160 / 3

h = 53.33 m

Conclusion

In this article, we used the angles of elevation and depression to calculate the height of a mountain. We used the tangent function to relate the angle of elevation, the height of the mast, and the height of the mountain. We solved for the height of the mountain and found that it is approximately 53.33m.

References

Trigonometry: A Unit Circle Approach, by Michael Corral
Calculus: Early Transcendentals, by James Stewart

Mathematical Formulas

tan(θ) = opposite side / adjacent side
sin(θ) = opposite side / hypotenuse
cos(θ) = adjacent side / hypotenuse

Glossary

Angle of elevation: the angle between the horizontal and the line of sight to the top of the object.
Angle of depression: the angle between the horizontal and the line of sight to the base of the object.
Trigonometry: the branch of mathematics that deals with the relationships between the sides and angles of triangles.
The Angle of Elevation and Depression of the Top and Base of a Mountain: Q&A ====================================================================

Introduction

In our previous article, we used the angles of elevation and depression to calculate the height of a mountain. In this article, we will answer some frequently asked questions related to the topic.

Q: What is the angle of elevation and depression?

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

Q: How do I calculate the height of a mountain using the angle of elevation and depression?

A: To calculate the height of a mountain, you need to use the concept of trigonometry. You can use the tangent function to relate the angle of elevation, the height of the mast, and the height of the mountain.

Q: What is the formula for calculating the height of a mountain?

A: The formula for calculating the height of a mountain is:

h = (x * tan(θ)) + h_mast

where h is the height of the mountain, x is the distance from the mast to the base of the mountain, θ is the angle of elevation, and h_mast is the height of the mast.

Q: How do I calculate the distance from the mast to the base of the mountain?

A: To calculate the distance from the mast to the base of the mountain, you can use the formula:

x = h_mast / tan(θ)

where x is the distance from the mast to the base of the mountain, h_mast is the height of the mast, and θ is the angle of depression.

Q: What are some common mistakes to avoid when calculating the height of a mountain?

A: Some common mistakes to avoid when calculating the height of a mountain include:

Not using the correct angle of elevation or depression
Not using the correct height of the mast
Not using the correct formula for calculating the height of the mountain
Not checking the units of measurement

Q: Can I use the angle of elevation and depression to calculate the height of a building or a tree?

A: Yes, you can use the angle of elevation and depression to calculate the height of a building or a tree. The formula for calculating the height of a building or a tree is the same as the formula for calculating the height of a mountain.

Q: How accurate is the calculation of the height of a mountain using the angle of elevation and depression?

A: The accuracy of the calculation of the height of a mountain using the angle of elevation and depression depends on the accuracy of the measurements and the angle of elevation and depression. If the measurements and angles are accurate, the calculation will be accurate.

Conclusion

In this article, we answered some frequently asked questions related to the topic of the angle of elevation and depression of the top and base of a mountain. We hope that this article has been helpful in understanding the concept of the angle of elevation and depression and how to calculate the height of a mountain using this concept.

References

Trigonometry: A Unit Circle Approach, by Michael Corral
Calculus: Early Transcendentals, by James Stewart

Mathematical Formulas

tan(θ) = opposite side / adjacent side
sin(θ) = opposite side / hypotenuse
cos(θ) = adjacent side / hypotenuse

Glossary

Angle of elevation: the angle between the horizontal and the line of sight to the top of the object.
Angle of depression: the angle between the horizontal and the line of sight to the base of the object.
Trigonometry: the branch of mathematics that deals with the relationships between the sides and angles of triangles.