A Surveyor's Trigonometry Dilemma: Finding the Height of a Building

The Problem:

A surveyor is 100 meters from the base of a building. The angle of elevation to the top of the building is 26°, The surveyor's instrument is 1.73 meters above the ground. Find the height of the building to the nearest tenth of a meter.

Final answer:

Using trigonometry, the tangent of the 26° angle of elevation is equal to the height of the building minus the height of the surveyor's instrument over the distance from the surveyor to the building base. Solving for the building's height and adding the height of the instrument gives approximately 50.8 meters.

Explanation:

To find the height of the building, we can use trigonometry. The problem provides the angle of elevation to the top of the building (26°) and the distance from the surveyor to the base of the building (100 meters). The surveyor's instrument height is 1.73 meters above the ground. The height of the building (h) can be found by using the tangent of the angle of elevation:

tangent(26°) = opposite/adjacent

So:

tangent(26°) = (h - 1.73 m) / 100 m

After solving for (h - 1.73 m), we get:

(h - 1.73 m) = 100 m * tangent(26°)

(h - 1.73 m) ≈ 49.1 m

Then:

h ≈ 49.1 m + 1.73 m

h ≈ 50.83 m

Since we want the answer to the nearest tenth of a meter:

h ≈ 50.8 meters