What is the length of the longest rod that can be placed in a room which is 3 metres long, 4 metres broad and 5 metres high ? (m metres)

Options

A. 5
B. 12
C. `5sqrt(2)`
D. `6sqrt(2)`

Answer

To find the length of the longest rod that can be placed in a room shaped like a cuboid, we can use the formula for the diagonal of a cuboid. The diagonal \( d \) can be calculated using the formula:

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

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

Given:
- Length \( l = 3 \) meters,
- Breadth \( b = 4 \) meters,
- Height \( h = 5 \) meters.

Now, we can substitute these values into the formula.

### Step 1: Square the dimensions
Calculate \( l^2 \), \( b^2 \), and \( h^2 \):
\[ l^2 = 3^2 = 9 \]
\[ b^2 = 4^2 = 16 \]
\[ h^2 = 5^2 = 25 \]

### Step 2: Add the squared values
Now, add these squared values together:
\[ l^2 + b^2 + h^2 = 9 + 16 + 25 = 50 \]

### Step 3: Take the square root
Finally, take the square root of the sum to find the length of the diagonal:
\[ d = \sqrt{50} \]
\[ d = \sqrt{25 \times 2} = 5\sqrt{2} \approx 7.07 \text{ meters} \]

Thus, the length of the longest rod that can be placed in the room is approximately \( 7.07 \) meters.

### Final Answer:
The length of the longest rod that can be placed in the room is \( 5\sqrt{2} \) meters or approximately \( 7.07 \) meters.

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