Calculating the Height of a Tower Using Trigonometry

The problem of determining the height of a tower using the angle of elevation from different points has been a classic application of trigonometry. This article will explore the problem of finding the height of a tower given the angle of elevation from the roof and the basement of a house. We will use the properties of triangles and trigonometric functions to solve this interesting mathematical challenge.

Problem Statement

The angle of elevation from the roof of the house to the top and bottom of the tower is 30° and 60°, respectively. The height of the house is 50 feet. The question is to find the height of the tower.

Given Information

The angle of elevation from the roof of the house (D) to the top of the tower (Q) is 60°. The angle of elevation from the basement of the house (B) to the top of the tower (Q) is 30°. The height of the house (AB) is 50 feet.

Step-by-Step Solution

Step 1: Define Variables

Let d be the distance from the house to the tower. Let t be the height of the tower.

Step 2: Use the Angle of Elevation to Form Equations

From roof (D) to the top of the tower:

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

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

Step 3: Solve for d

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

Step 4: Use the Angle of Elevation from the Basement to Form Another Equation

From basement (B) to the top of the tower:

(tan 30^circ frac{t - 50}{d})

(frac{1}{sqrt{3}} frac{t - 50}{d})

Step 5: Substitute d from Step 3 into the Equation from Step 4

(frac{1}{sqrt{3}} frac{t - 50}{frac{t}{sqrt{3}}})

(frac{1}{sqrt{3}} frac{t - 50}{t})

(t 3(t - 50))

(t 3t - 150)

(150 2t)

(t 75)

The height of the tower is 75 feet.

Verifying the Solution

Let's verify the solution by checking if the distance d from the house to the tower and the consistency of the angles of elevation.

From roof to the top of the tower:

(d frac{75}{sqrt{3}} 25sqrt{3} approx 43.3) feet

From basement to the top of the tower:

(frac{75 - 50}{43.3} frac{25}{43.3} frac{1}{sqrt{3}})

This confirms that the solution is correct.

Conclusion

The height of the tower, as calculated using trigonometric principles, is 75 feet. This calculation is a practical example of how trigonometry can be used to solve real-world problems involving heights and distances.

Keywords

angle of elevation, trigonometry, height calculation

References

[1] St,-.