A geologist stands on the shore of a lake in Ontario with a theodolite and finds that the angle of inclination to the top of a distant rock pillar is 28.5 degrees. He then walks 10 m along the shore at a 90 degree angle to his measurment, and finds that the angle to the same point on the bottom of the rock pillar is 56.4 degrees. Find the height of the rock pillar in metres, given that the theodolite was 1.6 metres above the ground.

Question

A geologist stands on the shore of a lake in Ontario with a theodolite and finds that the angle of inclination to the top of a distant rock pillar is 28.5 degrees. He then walks 10 m along the shore at a 90 degree angle to his measurment, and finds that the angle to the same point on the bottom of the rock pillar is 56.4 degrees. Find the height of the rock pillar in metres, given that the theodolite was 1.6 metres above the ground.

Answer 1

Is the 56.4º at the base of the rock or at the new position of the geologist?

I took it to be at the new position of the geologist.

The line from the base of the rock to the geologist's original position is common to two triangles.
I let the height of the rock be h m

then tan28.5 = h/x ---> x = h/tan28.5

and

cos 56.4 = 10/x ----> x = 10/cos 56.4

then h/tan28.5 = 10/cos56.4
h = 9.83

Assuming that he kept his theodolite reading horizontally to the base of the rock, let's add the 1.6 m to get
a height of 11.43 m