A mountain climber is at an altitude of 2.9 mi above the earth’s surface. From the climber’s viewpoint, what is the distance to the horizon? Enter your answer as a decimal in the box. Round only your final answer to the nearest tenth.
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: [tex]a^{2} + b^{2} = c^{2} [/tex]
Where: a = 3959 b = x c = 3959+2.9=3961.9 [tex] 3959^{2} + x^{2} = 3961.9^{2} \\ \\ x^{2} =3961.9^{2} - 3959^{2} \\ \\ x^{2} =22970.61 \\ \\ x=151.6[/tex]
From the climber's viewpoint the horizon is at distance of 151.6 miles.
