Question

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.

Answer 1

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.