Question

Mia is trying to find the height of a radio antenna on the roof of a local building. She stands at a horizontal distance of 15 meters from the building. The angle of elevation from her eyes to the roof left bracket(point AAright bracket) is 18degrees

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, and the angle of elevation from her eyes to the top of the antenna left bracket(point BBright bracket) is 53degrees
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. If her eyes are 1.75 meters from the ground, find the height of the antenna left bracket(the distance from point AA to point BBright bracket). Round your answer to the nearest tenth of a meter if necessary.

Answer 1

To determine the height of the antenna, trigonometry using the tangent function is applied to the angles of elevation and the known horizontal distance. The heights of the building and the antenna above eye level are calculated and their difference gives the height of the antenna.

Step-by-step explanation:

Mia wants to find the height of an antenna (point B) on top of a building given the angle of elevation to the top of the building (point A) is 18 degrees, and the angle of elevation to the top of the antenna is 53 degrees. The horizontal distance from Mia to the building is 15 meters, and her eye level is 1.75 meters above the ground.

To find the height of the antenna, we can use trigonometry, specifically the tangent function, which is the ratio of the opposite side to the adjacent side in a right triangle. For the triangle formed by point A, Mia's eye level, and the base of the building, the height of point A above Mia's eye level can be found by:

1. Calculating the height of the building up to point A using the tangent of 18 degrees: tan(18 degrees) = (Height of A - 1.75 m) / 15 m.
2. Finding the total height: (Height of A) = 15 m * tan(18 degrees) + 1.75 m.

Using a similar process, we calculate the height of the antenna (point B) as follows:

1. Calculating the height of point B above Mia's eye level using the tangent of 53 degrees: tan(53 degrees) = (Height of B - 1.75 m) / 15 m.
2. Finding the total height: (Height of B) = 15 m * tan(53 degrees) + 1.75 m.

The height of the antenna (the distance from point A to point B) is the difference between Height of B and Height of A. Round this final answer to the nearest tenth of a meter.