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Cubical Box | AMC-10A, 2010 | Problem 20

Try this beautiful problem from Geometry:cubical box from AMC-10A, 2010. You may use sequential hints to solve the problem

Try this beautiful problem from Geometry based on cubical box.

Cubical Box – AMC-10A, 2010- Problem 20


A fly trapped inside a cubical box with side length $1$ meter decides to relieve its boredom by visiting each corner of the box. It will begin and end in the same corner and visit each of the other corners exactly once. To get from a corner to any other corner, it will either fly or crawl in a straight line. What is the maximum possible length, in meters, of its path?

  • \(4\sqrt 5+3\sqrt 2\)
  • \(4\sqrt 5+3\sqrt 2\)
  • \(4\sqrt 3+4\sqrt 2\)
  • \(7\sqrt 3+4\sqrt 2\)
  • \(4\sqrt 3+7\sqrt 2\)

Key Concepts


Geometry

Cubical

Pythagoras

Check the Answer


Answer: \(4\sqrt 3+4\sqrt 2\)

AMC-10A (2010) Problem 20

Pre College Mathematics

Try with Hints


cube

Given that “A fly trapped inside a cubical box with side length \(1\) meter decides to relieve its boredom by visiting each corner of the box. It will begin and end in the same corner and visit each of the other corners exactly once”……………………………Therefore we may say that from a corner to any other corner the straight path will be \(A \to G \to B \to H \to C \to E \to D \to F \to A\)

can you finish the problem……..

cube

The distance of an interior diagonal in this cube is \(\sqrt 3\) ( i.e \(HB\)) and the distance of a diagonal on one of the square faces is \(\sqrt 2\) ( i.e \(HA\))

can you finish the problem……..

cubical box

Now the fly visits each corner exactly once, it cannot traverse such a line segment twice. Also, the cube has exactly four such diagonals, so the path of the fly can contain at most four segments of length.Therefore  the maximum distance traveled is \(4\sqrt 3+4\sqrt 2\)

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