Asurveyor measures the angle of elevation to a point on a mountain, that is 6 miles away, to be 15 degrees. the vertical change in the elevation from the point where the surveyor is standing to the point on the mountain is miles. (round your answer to the nearest hundredth of a mile)

You can draw a right triangle with the informatio.

The vertical change in elevation (unknown) is the opposed leg to the angle 15°

The distance from the surveyor to the point vertically below the point on the mountain is the adjacent leg = 6 miles.

Then tan(15°) = x / 6 => x = 6tan(15) = 1.61 miles

1.61 miles

The surveyor and mountain form a right angle triangle. 
The base of this triangle is 6 miles and the angle between its hypotenuse and base is 15°
Given that we need to find the perpendicular height, we can apply:
tan∅ = perpendicular/base
p = 6 x tan(15)
= 1.61 miles

What can we do about it

One that half its volume