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Minimizing a Path in the Coordinate plane

You start on point \(A=(0,0)\) and you want to get to point \(B=(10,1)\). There is a circular object with radius \(2\) blocking your way: it's equation is \((x-n)^2+y^2=4\) for some \(n\in [2,8]\). Let the shortest path from \(A\) to \(B\) such that you do not pass through the circular object have length \(P\). What should \(n\) be such that \(P\) is minimized?


Maybe surprisingly, the answer is not \(n=8\).

You can use Wolfram Alpha to bash it.

Diagram will be added ASAP.

Note by Daniel Liu
3 years, 7 months ago

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For n = 2, distance = 8 + pi, which I guess is the shortest distance.

Vineeth Chelur - 3 years, 7 months ago

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Not quite! Good try.

Since nobody has replied, I will give the answer: the shortest distance is approximately \(10.4583\) at \(n\approx 6.56261\).

Daniel Liu - 3 years, 7 months ago

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After working out for nearly 4 hours, I found a smaller value. The answer is 10.3597 at n = 6.87425. I will post the solution tomorrow. The answer will be a bit smaller than this answer because of calculator limit I had to approximate.

Vineeth Chelur - 3 years, 7 months ago

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