Find the length of the longest pole that can be placed in a room 30 m long, 24 m broad and 18 m high .

Answer

To find the length of the longest pole that can be placed in a room with dimensions 30 m (length), 24 m (breadth), and 18 m (height), we need to calculate the diagonal of the room. The diagonal can be found using the formula for the diagonal of a rectangular prism:

\[ d = \sqrt{l^2 + b^2 + h^2} \]

where:
- \(d\) is the diagonal,
- \(l\) is the length,
- \(b\) is the breadth,
- \(h\) is the height.

### Step-by-Step Solution:

1. **Identify the dimensions of the room**:
- Length (\(l\)) = 30 m
- Breadth (\(b\)) = 24 m
- Height (\(h\)) = 18 m

2. **Square each dimension**:
- \(l^2 = 30^2 = 900\)
- \(b^2 = 24^2 = 576\)
- \(h^2 = 18^2 = 324\)

3. **Add the squared dimensions**:
\[ l^2 + b^2 + h^2 = 900 + 576 + 324 \]
\[ = 1800 \]

4. **Take the square root of the sum**:
\[ d = \sqrt{1800} \]

5. **Simplify \(\sqrt{1800}\)**:
- Factor 1800: \(1800 = 100 \times 18 = 100 \times 9 \times 2 = 10^2 \times 3^2 \times 2\)
- Therefore, \(\sqrt{1800} = \sqrt{10^2 \times 3^2 \times 2} = 10 \times 3 \times \sqrt{2} = 30\sqrt{2}\)

6. **Final result**:
The length of the longest pole that can be placed in the room is \(30\sqrt{2}\) meters.