Calculating Building Height Using Angle of Elevation

This problem involves finding the height of a building using trigonometry, specifically the concept of the angle of elevation. We are given the distance from the foot of the building and the angle of elevation to the top of the building.

Understanding the Angle of Elevation Problem

The angle of elevation is the angle formed by the horizontal line of sight and the line of sight upwards to an object. In this case, the object is the top of the building. We can visualize this scenario as a right-angled triangle where:

• The height of the building is the side opposite the angle of elevation.
• The distance from the foot of the building is the side adjacent to the angle of elevation.
• The line of sight from the observer's eye to the top of the building is the hypotenuse.

Applying Trigonometry to Find Height

In a right-angled triangle, the trigonometric ratio that relates the opposite side and the adjacent side is the tangent function (tan). The formula is:

\(\tan(\text{angle}) = \frac{\text{Opposite side}}{\text{Adjacent side}}\)

In our problem:

• Angle of elevation = \(60^\circ\)
• Adjacent side (Distance from foot) = 70 m
• Opposite side (Height of the building) = Let's call it \(h\)

So, we can write the equation:

\(\tan(60^\circ) = \frac{h}{70}\)

Solving for the Height of the Building

To find the height \(h\), we need to multiply both sides of the equation by 70:

\(h = 70 \times \tan(60^\circ)\)

We know that the value of \(\tan(60^\circ)\) is \(\sqrt 3\).

Substituting this value into the equation:

\(h = 70 \times \sqrt 3\)

\(h = 70\sqrt 3 \text{ m}\)

Therefore, the height of the building is \(70\sqrt 3\) meters.

Step-by-Step Solution Summary

1. Identify the given information: angle of elevation (\(60^\circ\)), distance from the foot (70 m).
2. Recognize the problem forms a right-angled triangle.
3. Identify the required side (height) as the opposite side and the given distance as the adjacent side to the angle of elevation.
4. Choose the appropriate trigonometric ratio: tangent (\(\tan\)), which relates opposite and adjacent sides.
5. Set up the equation: \(\tan(60^\circ) = \frac{\text{Height}}{70}\).
6. Recall the value of \(\tan(60^\circ) = \sqrt 3\).
7. Solve for the height: Height = \(70 \times \tan(60^\circ) = 70 \times \sqrt 3 = 70\sqrt 3\) m.

Revision Table: Common Trigonometric Values

Additional Information: Angle of Elevation and Right Triangles

Angle of Elevation: This is an angle measured upwards from a horizontal line to a point above the observer. It's a key concept in solving problems involving heights and distances using trigonometry.

Right Triangles: A right-angled triangle is a triangle in which one of the angles is exactly \(90^\circ\). The sides of a right triangle have specific relationships defined by trigonometric ratios (sine, cosine, tangent), which are essential for solving problems like finding the height of a building or the distance to an object.