Nicole is trying to find the height of a radio antenna on the roof of a local building. She stands at a horizontal distance of 18 meters from the building. The angle of elevation from her eyes to the roof left parenthesis(point AAright parenthesis) is 41degrees ∘ , and the angle of elevation from her eyes to the top of the antenna left parenthesis(point BBright parenthesis) is 44degrees ∘ . If her eyes are 1.68 meters from the ground, find the height of the antenna left parenthesis(the distance from point AA to point BBright parenthesis). Round your answer to the nearest meter if necessary.
Question
Nicole is trying to find the height of a radio antenna on the roof of a local building. She stands at a horizontal distance of 18 meters from the building. The angle of elevation from her eyes to the roof left parenthesis(point AAright parenthesis) is 41degrees ∘ , and the angle of elevation from her eyes to the top of the antenna left parenthesis(point BBright parenthesis) is 44degrees ∘ . If her eyes are 1.68 meters from the ground, find the height of the antenna left parenthesis(the distance from point AA to point BBright parenthesis). Round your answer to the nearest meter if necessary.
Solution
To solve this problem, we can use the tangent of the angles of elevation, which is the ratio of the opposite side (height) to the adjacent side (distance from the building).
First, let's find the height of the building up to the roof (point A). We can use the tangent of the angle of elevation to the roof, which is 41 degrees.
tan(41) = height_A / 18 height_A = 18 * tan(41)
Next, let's find the total height up to the top of the antenna (point B) using the tangent of the angle of elevation to the antenna, which is 44 degrees.
tan(44) = height_B / 18 height_B = 18 * tan(44)
The height of the antenna is the difference between the total height (height_B) and the height of the building up to the roof (height_A). Also, we need to add the height of Nicole's eyes from the ground, which is 1.68 meters.
height_antenna = height_B - height_A + 1.68
Now, you can plug in the values into a calculator to find the numerical values for height_A, height_B, and height_antenna. Remember to round your answer to the nearest meter if necessary.
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