Question 1083403
Start off by drawing the problem out. Make sure you label everything as much as possible. This is one way to do it.
<img src = "https://i.imgur.com/EuoRiOe.png">
The points A through D are:
A = peak of the mountain where the climber is located
B = base of the mountain (directly below the climber)
C = center of the Earth
D = furthest point (on the horizon) that the climber can see


The segment lengths are:
AD = x
AB = 3.1
BC = 3959
CD = 3959


Using segments AB and BC, we can say
AC = AB + BC
AC = 3.1+3959
AC = 3962.1
assuming ABC is a straight angle.

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Since D is a tangent point, angle ADC is a right angle. Therefore, triangle ADC is a right triangle. 


Use the pythagorean theorem to solve for x


(AD)^2 + (CD)^2 = (AC)^2
(x)^2 + (3959)^2 = (3962.1)^2
x^2 + 15673681 = 15698236.41
x^2 + 15673681 - 15673681 = 15698236.41 - 15673681
x^2 = 24555.41
sqrt(x^2) = sqrt(24555.41)
x = 156.701659


The x value is approximate to 6 decimal places.


If you want to round to the nearest tenth, then the accuracy reduces to 1 decimal place. 


So we go from 156.701659 to 156.7


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Final Answer: <font color=red>156.7 miles</font>