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Question Number 45122 by MrW3 last updated on 09/Oct/18

Commented by MrW3 last updated on 09/Oct/18

A mountain has the shape of a semi  sphere with radius R. From a point A  at the foot of mountain to a point B   over point A two paths will be built.  One path should have the shortest  length. The other one should have  an uniform slope. Find the lengthes  of both paths in terms of R and θ.

$${A}\:{mountain}\:{has}\:{the}\:{shape}\:{of}\:{a}\:{semi} \\ $$$${sphere}\:{with}\:{radius}\:{R}.\:{From}\:{a}\:{point}\:{A} \\ $$$${at}\:{the}\:{foot}\:{of}\:{mountain}\:{to}\:{a}\:{point}\:{B}\: \\ $$$${over}\:{point}\:{A}\:{two}\:{paths}\:{will}\:{be}\:{built}. \\ $$$${One}\:{path}\:{should}\:{have}\:{the}\:{shortest} \\ $$$${length}.\:{The}\:{other}\:{one}\:{should}\:{have} \\ $$$${an}\:{uniform}\:{slope}.\:{Find}\:{the}\:{lengthes} \\ $$$${of}\:{both}\:{paths}\:{in}\:{terms}\:{of}\:{R}\:{and}\:\theta. \\ $$

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