A cubical room has dimensions ` 4 ft xx 4 ft xx 4 ft` . An insect starts from one corner O and reaches a corner on the opposite side of the body diagonal.
Suppose that insect does not fly but crawls. Find the minimum distance travelled by insect to reach the destination.

Options

A. ` (4sqrt( s)+4) ft`
B. ` sqrt ( 48 ) ft`
C. ` sqrt ( 80 ) ft`
D. None of these

Explanation

To find the minimum distance traveled by the insect from one corner of a cubical room to the opposite corner along the surfaces of the cube, we can analyze the problem step by step.

### Step 1: Understand the Cube Dimensions
The dimensions of the cube are given as \(4 \, \text{ft} \times 4 \, \text{ft} \times 4 \, \text{ft}\). The insect starts at one corner (let's call it point O) and needs to reach the opposite corner (let's call it point A).

### Step 2: Identify the Path Options
The insect can only crawl along the surfaces of the cube. There are several possible paths the insect can take to reach point A. We will consider three potential paths:

1. **Path 1**: Move along three edges of the cube.
2. **Path 2**: Move along two edges and then diagonally across one face of the cube.
3. **Path 3**: Move diagonally across two faces of the cube.

### Step 3: Calculate the Distance for Each Path

#### Path 1: Three Edges
The distance traveled along three edges is simply the sum of the lengths of the edges:
\[ \text{Distance} = 4 + 4 + 4 = 12 \, \text{ft} \]

#### Path 2: Two Edges and One Face Diagonal
For this path, the insect travels along two edges (4 ft each) and then diagonally across the face of the cube. The diagonal distance can be calculated using the Pythagorean theorem:
\[ \text{Diagonal} = \sqrt{(4^2 + 4^2)} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \, \text{ft} \]
Thus, the total distance for this path is:
\[ \text{Distance} = 4 + 4 + 4\sqrt{2} \approx 8 + 5.66 \approx 13.66 \, \text{ft} \]

#### Path 3: Diagonal Across Two Faces
For this path, the insect crawls diagonally across two adjacent faces. The distance can be calculated as follows:
1. The insect moves 4 ft along one edge.
2. It then moves diagonally across the face, which is the hypotenuse of a right triangle with legs of 4 ft and 2 ft (half of the cube's edge):
\[ \text{Diagonal} = \sqrt{(4^2 + 2^2)} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \, \text{ft} \]
Thus, the total distance for this path is:
\[ \text{Distance} = 4 + 2\sqrt{5} \approx 4 + 4.47 \approx 8.47 \, \text{ft} \]

### Step 4: Compare the Distances
Now we compare the distances calculated for each path:
- Path 1: 12 ft
- Path 2: Approximately 13.66 ft
- Path 3: Approximately 8.47 ft

### Conclusion
The minimum distance traveled by the insect to reach the opposite corner of the cube along the surfaces is:
\[ \text{Minimum Distance} = 8.47 \, \text{ft} \]