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How tall would a tower at the South Pole have to be for the top to get year-around sunshine?

No doubt the height would vary each year since the Tropic of Cancer changes latitude slightly. Perhaps assume 23.5° N for calculation's sake.

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assuming spherical earth with radius $R$ and axis tilt $\mu$, from geometry reasoning we get: $$ \frac{R}{R+h}=\cos \mu \Rightarrow R\left( \frac{1}{\cos \mu} - 1 \right)=h $$ earth

so your tower height should be $h=((1/\cos(23.5)-1)\cdot 6371 \text{ km} = 576.2 \text{ km}$; greetings to ISS

for ellipsoid-Earth the height is bit different, but the above is a fine estimate

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    $\begingroup$ +1 for drawing the south pole at the top of the image contrary to arbitrary convention ;-) $\endgroup$ – joriki Aug 17 '11 at 7:05
  • $\begingroup$ "greetings to ISS" - I missed this at first reading... :D $\endgroup$ – J. M. isn't a mathematician Aug 17 '11 at 7:15
  • $\begingroup$ lol, yeah, I probably put myself too much in the Antarctic... $\endgroup$ – IljaBek Aug 17 '11 at 7:15

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