Brooklyn is trying to find the height of a radio antenna on the roof of a local building. She stands at a horizontal distance of 23 meters from the building. The angle of elevation from her eyes to the roof ((point AA)) is 24^{\circ} ∘ , and the angle of elevation from her eyes to the top of the antenna ((point BB)) is 38^{\circ} ∘ . If her eyes are 1.66 meters from the ground, find the height of the antenna ((the distance from point AA to point BB)). Round your answer to the nearest meter if necessary.

Trigonometric ratios indicates the ratio of the sides of a right triangle, and have values based on the lengths of the sides of a right triangle, drawn on the coordinate plane.

The distance from the building at which Brooklyn stands = 23 meters

Angle of elevation from her eyes to the roof, θ = 24°

Angle of elevation from her eyes to the top of the antenna = 38°

Height of her eyes from the ground = 1.66 meters

[tex]tan(\theta) = \dfrac{Opposite}{Adjacent} = \dfrac{Height}{Horizontal \, distance}[/tex]

Therefore;

Height = Horizontal distance × tan(θ)

Height to the top of the roof, H₁ = Horizontal distance × tan(θ)

Therefore;

H₁ = 23 × tan(24°) ≈ 10.24

Height to the top of the antenna, H₂ = 23 × tan(38°) ≈ 17.97

Height of the antenna ≈ 17.97 - 10.24 ≈ 8

Height of the antenna is about 8 meters

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