Question

A non-flying ant wants to travel from the bottom corner to the diagonally opposite top corner of a cubical room. The side of the room is 2 meters. What will be the minimum distance that the ant needs to travel?

• A. 6 meters
• B. 2√2+2 meters
• C. 2√3 meters
• D. 2√6 meters
• E. 2√5 meters

Answer 1

Answer: (E)

The cube has side length = 2 m. Start point: \( A(0,0,0) \) End point: \( C(2,2,2) \).

Case 1: Through the cube (not allowed)
Interior diagonal distance: \[ D = \sqrt{(2-0)^2 + (2-0)^2 + (2-0)^2} = \sqrt{4+4+4} = \sqrt{12} = 2\sqrt{3}. \] But the ant cannot move through the interior.

Case 2: On the surface of the cube
Unfold the cube into a net. The shortest surface path is the hypotenuse of a rectangle with dimensions: \[ \text{one side } = 2, \quad \text{other side } = 2+2 = 4. \] Distance: \[ H = \sqrt{2^2 + 4^2} = \sqrt{4+16} = \sqrt{20} = 2\sqrt{5}. \]

Final Answer:
The minimum distance the ant needs to crawl on the cube’s surface is: \[ \boxed{2\sqrt{5} \ \text{meters}} \]