To find the minimum distance that the insect must crawl to reach the opposite corner of a cubical room with dimensions \(4 \, \text{ft} \times 4 \, \text{ft} \times 4 \, \text{ft}\), we can visualize the problem in a 3D space and consider the different paths the insect can take.
### Step-by-Step Solution:
1. **Identify the corners**:
- Let the starting corner \(O\) be at coordinates \((0, 0, 0)\).
- The opposite corner \(A\) will then be at coordinates \((4, 4, 4)\).
2. **Consider the paths**:
- The insect can crawl along the edges of the cube, or it can take a diagonal path across the faces of the cube. We will explore the minimum distance by unfolding the cube.
3. **Unfolding the cube**:
- One way to visualize the shortest path is to unfold the cube into a 2D plane. The insect can crawl along the surfaces of the cube.
- For example, if we unfold the cube such that two adjacent faces are laid flat, we can create a rectangle that represents the path.
4. **Calculate the distance**:
- One of the shortest paths can be represented on the unfolded cube as a straight line from point \(O\) to point \(A\).
- If we consider the insect crawling from \((0, 0, 0)\) to \((4, 4, 4)\) through the unfolded path, the coordinates can be represented as:
- From \((0, 0)\) to \((4, 4)\) on a 2D plane.
- The distance \(d\) can be calculated using the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
- Here, \(x_1 = 0\), \(y_1 = 0\), \(x_2 = 4\), and \(y_2 = 4\):
\[
d = \sqrt{(4 - 0)^2 + (4 - 0)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \, \text{ft}
\]
5. **Conclusion**:
- The minimum distance travelled by the insect to reach the opposite corner is \(4\sqrt{2} \, \text{ft}\).
