How Vast Is Your Vision?

Let us assume the scenario to be like this.

The line of sight of the man, assuming perfectly ideal vision, will be perpendicular to the surface of the Earth, as the horizon serves as its point of tangency.

Thus, the line of sight of the man in all directions define the set of points tangent to the sphere (which we perceive as the horizon), which will be a spherical cap whose edge is perpendicular to the straight eye level line of sight of the man. The area enclosed by this is depicted in the figure.

The area of a spherical cap may be derived by using the formula for surface area integrals for sphere sections. From there, we will use this

$S \: = \: 2 \pi R^2 ( \: 1 \: - \: cos \: \theta)$ ..

If we cut this planet crosswise, we will be able to visualize this.

Now, we can easily get $\theta$ by using the general ratio for right triangles (remember that the line of sight is perpendicular to a radius of a sphere). This gives us

$cos \: \theta \: = \: \frac {R}{R\:+\:h}$

where $h$ represents the combined height of the mountain and the man's eye to ground level.

From this we can simplify the formula for $S$ .

$S \: = \: 2 \pi R^2 ( \: 1 \: - \frac {R}{R\:+\:h})$

$S \: = \: 2 \pi R^2 ( \frac {h}{R\:+\:h} )$

which means we only need to find out

$\: 2 ( \frac {h}{R\:+\:h} )$

given $h\: =\: 8.85$ km, and $R\: = \: 6371$ km, we find the ratio to be

$\frac {17.70}{6379.85} = \frac {354}{127597}$

That will ultimately give us $354 \: +\: 127597 \: = \: \boxed {127951}$ .

Sorry for the poor drawings.

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Note: I wasn't sure of the formula for the surface area of a spherical cap at first, so I tried to derive one from this integral

$A\: =\:2 \pi \large \int_{r-h}^r x \sqrt { 1 + (y')^2 } dx$

and got $A\: = \: 2 \pi R h$

well the $h$ here had nothing to do with the variable $h$ mentioned and used above. $h$ here refers to the maximum height of the spherical cap. Well, $h$ here is equal to $R\: - R\: cos\: \theta$ . That just justifies the formula used above.