A hall has the dimensions 10m `xx` 10m `xx` 10 m. A fly starting at one corner ends up at a diagonally opposite corner. The magnitude of its displacement is nearly

Options

A. `5sqrt2m `
B. ` 10sqrt3m`
C. `20sqrt3m`
D. `30sqrt3m`

Answer

To find the magnitude of the displacement of the fly that starts at one corner of a cube and ends up at the diagonally opposite corner, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
- We have a cube (hall) with dimensions 10m x 10m x 10m.
- The fly starts at one corner (let's call it point A) and ends at the diagonally opposite corner (point B).

2. **Identify the Coordinates**:
- Let's assign coordinates to the corners of the cube:
- Point A (0, 0, 0) - starting corner
- Point B (10, 10, 10) - diagonally opposite corner

3. **Use the Displacement Formula**:
- The displacement vector can be calculated using the distance formula in three-dimensional space:
\[ \text{Displacement} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
- Here, \((x_1, y_1, z_1)\) are the coordinates of point A and \((x_2, y_2, z_2)\) are the coordinates of point B.

4. **Substitute the Coordinates**:
- Substitute the coordinates of points A and B into the formula:
\[ \text{Displacement} = \sqrt{(10 - 0)^2 + (10 - 0)^2 + (10 - 0)^2} \]

5. **Calculate Each Term**:
- Calculate the squares:
\[ = \sqrt{10^2 + 10^2 + 10^2} = \sqrt{100 + 100 + 100} = \sqrt{300} \]

6. **Simplify the Expression**:
- The square root of 300 can be simplified:
\[ \sqrt{300} = \sqrt{100 \times 3} = 10\sqrt{3} \]

7. **Final Result**:
- Therefore, the magnitude of the displacement of the fly is:
\[ \text{Displacement} \approx 10\sqrt{3} \text{ meters} \]