A Classic Construction Problem

Another week, another Riddler. The question:

Consider four towns arranged to form the corners of a square, where each side is 10 miles long. You own a road-building company. The state has offered you $28 million to construct a road system linking all four towns in some way, and it costs you $1 million to build one mile of road. Can you turn a profit if you take the job?

Extra credit: How does your business calculus change if there were five towns arranged as a pentagon? Six as a hexagon? Etc.?

After a few napkin drawings I landed on this shape:

shape.png

Then I set up this series of equations:

Untitled2.png

Plugging that into Python we get answer of 27.3205 miles of bridges with an x_1 distance of 2.887.

Figure_1.png

Pretty simple but fun!

 

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