A hot air balloon is observed from two points A and B on the ground A and B are 800 feet apart the balloon is east of both points the angle of elevation from A to the balloon is 37 degrees and the angle of elevation from B is 65 degrees what is the distance from point A to the balloon?

Answer 1

The usual question for the above would be to find the height of the balloon, but this is easier

Hope you made a sketch.
Mine has a horizontal line PAB where the balloon is above P at a point Q
AB = 800
angle B = 37°
angle QBP = 65° , so angle QBA = 115°
which makes angle BQA = 180-115-37 = 28

by the sine law:
AQ/sin115 = 800/sin28°
AQ = 800sin115/sin28 = 1544.39 or 1544 to the nearest foot