How tall is the antenna?

Question

A broadcast antenna is located at the top of a building 1,000 feet tall.  From a point on the same horizontal plane as the base of the building, the angles of elevation of the top and bottom of the antenna are 65.8 degrees and 62.6 degrees respectively.  How tall is the antenna?

Answer 1

1. the angle of elevation formed by the ground and the bottom of the antenna is 62.6°

2. the side opposite to this angle is 1000

3. to find the distance from the building to the point of observation, we do: tan(62.6°) = 1000/a

4. a = 1000/tan(62.6°) ≈ 518.351

5. the angle of elevation formed by the ground and the top of the antenna is 65.8°

6. the side opposite to this angle is 1000 + the height of the antenna

7. to find this combined height, we do: tan(65.8°) = o/518.351

8. o = 518.351*tan(65.8°) ≈ 1153.383

9. now, let y = the height of the antenna

10. therefore, the height of the building + the height of the antenna = 1000 + y = 1153.383

11. subtract 1000 from both sides: y = 153.383

12. thus, the height of the antenna is approx. 153 ft