Shortest path...

Calculus Level 4

A road is to be constructed between four towns situated in such a way they form square. So, what is the minimum length of path joining these towns? Given that length between two adjacent towns is 1 km , i.e. side of square is 1 km.

None of these 2 2 2 \sqrt{2} Km 4 Km 1 + 3 1 + \sqrt{3} Km Can't say 2 \sqrt{2} Km

Sachin Vishwakarma by Sachin Vishwakarma , 22, India

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1 solution

Ivan Koswara
Jan 8, 2016

This is the Steiner tree problem on the vertices of the square. It's well known that the answer is 1 + 3 1+\sqrt{3} times the side length of the square (the network can be seen on the Wikipedia page; the internal vertices have angles 12 0 ∘ 120^\circ everywhere).

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