Giving many points!
in the center of a growing city are an old church and a high-rise office building separated by a plaza. two surveyors are collaborating on the job of determining how high the tip of the church spire is above the plaza. each surveyor has a theodolite mounted on a tripod. one surveyor is in the entrance way of the high-rise. the top of his tripod is 165cm above the plaza level. the second surveyor is directly above the first, and (amazingly) the top of her tripod is exactly 100m above the top of her colleague's tripod. the first surveyor measures the angle of elevation of the tip of the spire as 50∘. the second surveyor measures the angle of depression of the tip of the spire as 30∘.

correct to 2 decimal places, how high is the tip of the spire above the plaza?

  69.01 m

Step-by-step explanation:

The mnemonic SOH CAH TOA reminds you ...

  Tan = Opposite/Adjacent

The tangent function is useful for problems like this. Let the height of the spire be represented by h. The distance (d) across the plaza from the first surveyor satisfies the relation ...

  tan(50°) = (h -1.65)/d

Rearranging to solve for d, we have ...

  d = (h -1.65)/tan(50°)

The distance across the plaza from the second surveyor satisfies the relation ...

  tan(30°) = (101.65 -h)/d

Rearranging this, we have ...

  d = (101.65 -h)/tan(30°)

Equating these expressions for d, we can solve for h.

  (h -1.65)/tan(50°) = (101.65 -h)/tan(30°)

  h(1/tan(50°) +1/tan(30°)) = 101.65/tan(30°) +1.65/tan(50°)

We can divide by the coefficient of h and simplify to get ...

  h = (101.65·tan(50°) +1.65·tan(30°))/(tan(30°) +tan(50°))

  h ≈ 69.0148 ≈ 69.01 . . . . meters

The tip of the spire is 69.01 m above the plaza.