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.

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.

To determine the height of the hill (H), we can use trigonometry and the given information.

First, let's define the variables:
- H = height of the hill
- h = change in height from the bottom to the top of the hill
- H_I = height of the instrument
- A = height of the object
- Z = zenith angle
- s = horizontal distance between the surveying instrument and the object

Now, we can use the tangent function to find h:
tan(Z) = h / s

In this case, Z is given as 54 degrees 16 minutes 25 seconds. We need to convert this into decimal degrees for the calculation.

To convert the minutes and seconds to decimal degrees, we divide the minutes by 60 and the seconds by 3600, then add these values to the degrees.

Here's the calculation:
Z_decimal = 54 + 16/60 + 25/3600

Now we can substitute the values into the equation:
tan(Z_decimal) = h / s

We can rearrange the equation to solve for h:
h = tan(Z_decimal) * s

Substitute the given values:
h = tan(Z_decimal) * 120.30 ft

Now, we can calculate h using a scientific calculator. The equation can be written as follows:

h = tan(54.2736) * 120.30 ft

Calculate the value of tan(54.2736) using a scientific calculator.
In this case, tan(54.2736) is approximately 1.4473.

Substitute this value into the equation:
h ≈ 1.4473 * 120.30 ft

Calculating h:
h ≈ 174.38 ft

Therefore, the change in height from the bottom to the top of the hill (h) is approximately 174.38 ft.

To find the height of the hill (H), we need to add the height of the instrument (H_I) to h:
H = H_I + h

Substitute the given values:
H = 4.5 ft + 174.38 ft

Calculating H:
H ≈ 178.88 ft

Therefore, the height of the hill (H) is approximately 178.88 ft.