Asurveyor measures the angle of elevation to a point on a mountain to be 18 degrees the point on the mountain is horizontally 4 miles away from the surveyor the vertical change in elevation from the point where the surveyor is standing to the point on the mountain miles?

This is right triangle trig and you are looking for the height of the triangle. The reference angle is 18 degrees, the side adjacent to that angle is 4 and you are looking for the side opposite the reference angle.  The trig identity that helps us find that missing side is tangent:

and 4 tan(18) = x.
x = 1.3 miles