a television antenna is on the roof of the building. from a pt. on the ground 36.0 feet from the building , the angle of elevation of the top and the bottom of the antenna are 51 degree celsiusand 42 degree celsuis respectively. how tall is antenna?
tan42 = h1/36.
h1 =36*tan42 = 32.4 Ft. = Ht. of Bldg.
tan51 = (h1+h2)/36.
h1+h2 = 36*tan51 = 44.5 Ft. = Ht. of Bldg. plus ant.
hi + h2 = 44.5
h2 = 44.5 - h1 = 44.5 - 32.4 = 12.1 Ft. = Ht. of ant.
To find the height of the antenna, we can use trigonometric ratios and the given angles of elevation. Let's refer to the height of the antenna as "h".
From the given information, we can form a right triangle with the antenna, the ground point, and the top of the antenna as the three vertices. The position of the observer on the ground forms a line parallel to the ground, creating another right triangle with the antenna.
In the first triangle, the top angle is 51 degrees and the base of the triangle is the distance from the building to the observer, which is 36.0 feet. The height of the antenna is the opposite side in this triangle.
In the second triangle, the bottom angle is 42 degrees, the base is again the distance from the building to the observer (36.0 feet), and the height of the antenna is the adjacent side.
Using the tangent function, we can write the following equations:
tan(51) = h / 36.0 (equation 1)
tan(42) = h / 36.0 (equation 2)
To solve these equations simultaneously, we can isolate the height "h" by multiplying both sides of each equation by 36.0:
h = 36.0 * tan(51) (equation 3)
h = 36.0 * tan(42) (equation 4)
Now we can calculate the height:
h = 36.0 * tan(51) ≈ 41.24 feet
h = 36.0 * tan(42) ≈ 33.16 feet
Therefore, the height of the antenna is approximately 41.24 feet.