A hot air balloon is spotted simultaneously by a man and a woman 3 ½ miles apart. If the angle of elevation from the man to the balloon is 27° and the angle of elevation from the woman to the balloon is 41°, find the altitude of the balloon in feet.
+129840 A hot air balloon is spotted simultaneously by a man and a woman 3 ½ miles apart. If the angle of elevation from the man to the balloon is 27° and the angle of elevation from the woman to the balloon is 41°, find the altitude of the balloon in feet.
Let the distance that the man is from the ballon = x miles....then the woman is 3.5 - x miles from the balloon
And let h be the balloon height.....
From the man's point of view we have :
tan ( 27°) = h / x → x = h / tan ( 27°)
From the woman's point of view, we have
tan (41°) = h / [3.5 - x] substitute for x
tan ( 41°) = h / [ 3.5 - h / tan (27°) ] rearrange
tan (41°) [ 3.5 - h / tan (27°) ] = h distribute acroos the brackets
3.5 *tan (41°) - h ( tan41°) / tan (27°) = h add h ( tan41°) / tan (27°) to both sides
3.5 tan (41°) = h + h ( tan41°) / tan (27°) factor h out of the right side
3.5 tan (41°) = h [ 1 + ( tan41°) / tan (27°) ] divide both sides by [ 1 + ( tan41°) / tan (27°) ]
3.5 tan (41°) / [ 1 + ( tan41°) / tan (27°) ] = h ≈ 1.124 miles
