Calculating the Height of a Building Using Trigonometry

In this article, we will explore how to calculate the height of a building using the angles of elevation and trigonometric relationships. We'll solve a specific problem step-by-step and explain the mathematical concepts involved.

Problem Statement

The angle of elevation of the top of the building from the foot of the tower is 30 degrees. The angle of elevation of the top of the tower from the foot of the building is 60 degrees. If the tower is 60 m high, what is the height of the building?

Solution

To solve this problem, we will use trigonometric relationships involving the tangent function. Let's break it down into straightforward steps.

Step 1: Find the Distance Between the Building and the Tower

The first step is to use the given angle of elevation from the foot of the building to the top of the tower.

Given: The height of the tower ( h_t 60 ) m. The angle of elevation from the foot of the building to the top of the tower ( theta_t 60^circ ).

The relationship between the height and the distance using the tangent function is:

tan ( theta_t ) ( frac{h_t}{d} )

Substituting the known values:

tan ( 60^circ ) ( frac{60}{d} )

Since tan ( 60^circ ) ( sqrt{3} ):

( sqrt{3} frac{60}{d} )

Rearranging gives:

( d frac{60}{sqrt{3}} 20sqrt{3} ) m

Step 2: Calculate the Height of the Building

The next step is to use the angle of elevation from the foot of the tower to the top of the building.

Given: The angle of elevation from the foot of the tower to the top of the building ( theta_b 30^circ ). The distance between the building and the tower ( d 20sqrt{3} ) m.

The relationship between the height and the distance using the tangent function is:

tan ( theta_b ) (frac{h_b}{d})

Substituting the known values:

tan ( 30^circ ) (frac{h_b}{20sqrt{3}})

Since tan ( 30^circ ) (frac{1}{sqrt{3}}):

(frac{1}{sqrt{3}} frac{h_b}{20sqrt{3}})

Cross-multiplying gives:

( h_b 20sqrt{3} cdot frac{1}{sqrt{3}} 20 ) m

Conclusion

The height of the building is 20 meters.

Alternative Approach and Further Analysis

Alternatively, we can also assume the height of the tower from a person's eye level is h meters. The person is at a distance l1 from the tower when the elevation is 30 degrees and at a distance l2 when the elevation is 60 degrees.

Given: l1 - l2 10 m. tan 30 h/l1 1/√3. tan 60 h/l2 √3.

Using the tangent relationships:

l1 h√3 and l2 h/√3.

From the given l1 - l2 10 m:

10√3 3h - h 2h.

Therefore:

h 5√3 m ≈ 8.66 m.

So, the height of the tower from the person's eye level is approximately 8.66 meters.

Additional Insights

In the alternative approach, we see that the height of the building (assuming the tower's height from eye level) is calculated to be:

30 degrees / 30 degrees 3 / 1. The tower is 3 times as tall as the building. Given the tower's height is 60 m, the building's height calculated by this method is approximately 20 m, consistent with our initial solution.

Therefore, the height of the building is 20 meters.