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 by an ant to reach the farthest corner of a cubical room of side \( a \) while moving with a constant speed \( u \), we can follow these steps:

### Step 1: Understand the Geometry of the Problem
The ant starts at one corner of the cube (let's call it point A) and needs to reach the opposite corner (point B). The distance between these two points is the diagonal of the cube.

### Step 2: Calculate the Distance
The distance \( d \) between two opposite corners of a cube can be calculated using the 3D distance formula. For a cube with side length \( a \), the distance \( d \) is given by:
\[ d = \sqrt{a^2 + a^2 + a^2} = \sqrt{3a^2} = a\sqrt{3} \]

### Step 3: Relate Distance to Time
The time \( t \) taken to cover a distance \( d \) at a constant speed \( u \) can be expressed using the formula:
\[ t = \frac{d}{u} \]

### Step 4: Substitute the Distance
Substituting the expression for \( d \) into the time formula gives:
\[ t = \frac{a\sqrt{3}}{u} \]

### Step 5: Conclusion
Thus, the minimum time \( t \) taken by the ant to reach the farthest corner of the cube is:
\[ t = \frac{a\sqrt{3}}{u} \]