Posted by Teresa on .
a surveyor is trying to determine the change in hight h from the bottom to the top of a hill. If the height of instrument H is 4.5 ft the height of the object that he sights on the tower A is 6.5 ft and the zenith angle Z is 54deg 16' 25" determine the height of H (the hill) Assume that the horizontal distance (s) between the surveying instrument and the object (s) is 120.30 ft.

Geodetic Survey 
MathMate,
Unless you're very experienced in the area, it would simplify your thought process by drawing a diagram as described by the question.
Here is more or less what you should have:
http://imageshack.us/a/img441/9503/1347673609.jpg
You will need to solve for the height H using the tangent/cotangent ratio.
The difference between the bottom and top of the hill is therefore
H + 4.5  6.5 feet.