Amari is standing 50 feet from the base of a building. From where he stands, the angle formed between the top of the building and the ground at his feet is [tex]$60^{\circ}$[/tex]. How tall is the building?
A. 50 ft
B. [tex]$\frac{50 \sqrt{3}}{3} ft$[/tex]
C. [tex]$50 \sqrt{3} ft$[/tex]
D. 100 ft
Answer (1)
We have a right triangle where the height of the building is opposite to the 6 0 ∘ angle, and the distance from Amari to the building is the adjacent side.
Using the tangent function, we set up the equation: tan ( 6 0 ∘ ) = 50 h .
Since tan ( 6 0 ∘ ) = 3 , we have 3 = 50 h .
Solving for h , we find the height of the building: 50 3 f t .
Explanation
Problem Analysis Let's analyze the problem. We have a right triangle formed by Amari, the base of the building, and the top of the building. The distance from Amari to the base of the building is 50 feet, and the angle of elevation to the top of the building is 6 0 ∘ . We need to find the height of the building.
Applying Tangent Function We can use the tangent function to relate the angle of elevation, the distance from the building, and the height of the building. Let h be the height of the building. Then, we have: tan ( 6 0 ∘ ) = 50 h
Substituting the Value of Tangent We know that tan ( 6 0 ∘ ) = 3 . So, we can substitute this value into the equation: 3 = 50 h
Solving for h Now, we can solve for h by multiplying both sides of the equation by 50: h = 50 3
Final Answer Therefore, the height of the building is 50 3 feet.
Examples
Understanding angles of elevation is crucial in various real-world scenarios. For instance, architects use these angles to design buildings and structures, ensuring stability and proper alignment. Similarly, surveyors use angles of elevation to measure the heights of mountains or the depths of valleys. In navigation, sailors and pilots rely on angles of elevation to determine their position and direction, especially when using landmarks or celestial objects as reference points. Knowing how to calculate heights using angles of elevation helps in urban planning for skyscrapers, bridges, and other infrastructure projects, ensuring safety and precision in construction.
