In the figure below, the radius, AB=AD=3959 mi, the climber is at position C. BC=3.1 mi. CD=x mi, is the distance from the climber to the horizon. Thus to solve for x we proceed as follows:
AC=3959+3.1=3962.1 mi
To evaluate for x we use the Pythagorean theorem, this is given by:
c^2=a^2+b^2
c=AC is the hypotenuse
a and b are the legs
plugging in the values we shall have:
3962.1^2=x^2+3959^2
x^2=3961.1^2-3959^2
x^2=24555.41
hence
x=156.7 mi
Answer: 156.7 mi
