An ant is at a corner of a cubical room of side a. The ant can move with a constant speed u. The minimum time taken to reach the farthest corner of the cube is

Options

A. `(3a)/u`
B. `(sqrt3 a)/u`
C. `(sqrt5 a)/u`
D. `((sqrt2+1)a)/u`

Explanation

To solve the problem of finding the minimum time taken for an ant to travel from one corner of a cubical room to the farthest corner, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- The ant starts at one corner of a cube and needs to reach the opposite corner. The cube has a side length of \( a \).

2. **Identifying the Corners**:
- Let's denote the starting corner as point \( P \) and the farthest corner as point \( R \).

3. **Visualizing the Cube**:
- If we visualize or unfold the cube, we can see that the shortest path from \( P \) to \( R \) can be represented as a straight line across the unfolded surface of the cube.

4. **Finding the Shortest Path**:
- The distance between the two corners can be calculated using the Pythagorean theorem.
- The straight-line distance \( d \) from point \( P \) to point \( R \) can be represented as the hypotenuse of a right triangle formed by two edges of the cube.
- The distance can be calculated as:
\[ d = \sqrt{(a^2 + a^2 + a^2)} = \sqrt{3a^2} = a\sqrt{3} \]

5. **Calculating the Minimum Time**:
- The time \( t \) taken to travel this distance at a constant speed \( u \) is given by the formula:
\[ t = \frac{\text{distance}}{\text{speed}} = \frac{d}{u} \]
- Substituting the distance we found:
\[ t = \frac{a\sqrt{3}}{u} \]

6. **Final Result**:
- Therefore, the minimum time taken for the ant to reach the farthest corner of the cube is:
\[ t = \frac{a\sqrt{3}}{u} \]

### Conclusion:
The minimum time taken for the ant to reach the farthest corner of the cube is \( \frac{a\sqrt{3}}{u} \).