When climber is looking at horizon we can consider that he is looking straight at one point. We need to calculate length of a line extending from climber to horizon. When applied to a circle this line is called tangent. Tangent is a straight line that touches circle in just one point. By definition tangent is perpendicular to a radius of circle. This will help us to solve this problem.
From the picture we can see that we have a triangle. It is right angle triangle with right angle positioned at location where tangent intercepts radius. We can use the pythagorean theorem to solve this problem:
Where: a = 3959 b = x c = 3959+2.9=3961.9
From the climber's viewpoint the horizon is at distance of 151.6 miles.