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)

To find the minimum distance the ant needs to travel from the bottom corner to the diagonally opposite top corner of a cubical room, we can utilize the concept of unfolding the cube into a net.

Given that the side of the cube is 2 meters, let's understand the scenario better with a step-by-step explanation:

1. The ant needs to travel from the corner of a cube at point \( A \) (0,0,0) to its diagonally opposite corner \( B \) (2,2,2).
2. In a 3D space, the straight-line distance (diagonal of the cube) using the distance formula would be given by: \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}\).
3. Plugging the points A and B into the formula, we get: \(d = \sqrt{(2-0)^2 + (2-0)^2 + (2-0)^2} = \sqrt{4 + 4 + 4} = \sqrt{12} = 2\sqrt{3}\) meters.
4. However, since the ant is non-flying, it cannot travel through space but must crawl along the surfaces of the cube.
5. Consider "unfolding" the cube such that a path along the surfaces may be found in two dimensions. One possible unfolding is a net shaped like a T.
6. On the net, the shortest path is a straight line from the bottom corner to the top corner. This line goes across two squares of the cube net—one on the base and one on a side wall.
7. The net can be visualized as the two squares adjacent along the same vertical edge. From below diagonally moving through these two squares, the path becomes simple.
8. This follows directly through two adjacent squares in which each dimension is 2 meters, thus the path becomes: \(d = \sqrt{2^2 + 2^2 + 2^2} = \sqrt{4 + 4 + 4} = \sqrt{12} = 2\sqrt{3} = 2\sqrt{5}\) meters.

Therefore, the minimum distance the ant needs to travel along the surface to move from the bottom to the top opposite corner is 2√5 meters.

Answer 2

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: \[ \boxed{2\sqrt{5} \ \text{meters}} \]