What is the length of the longest rod that can be placed in a room which is 6 metres long, 8 metres broad and 20 metres high?

Options

A. `15sqrt5`
B. 20
C. 15
D. `10sqrt5`

Answer

To find the length of the longest rod that can be placed in a room with dimensions 6 meters long, 8 meters broad, and 20 meters high, we need to calculate the diagonal of the rectangular prism (the room). The formula for the diagonal \(D\) of a rectangular prism is given by:

\[ D = \sqrt{L^2 + B^2 + H^2} \]

where:
- \(L\) is the length of the room,
- \(B\) is the breadth (width) of the room,
- \(H\) is the height of the room.

### Step 1: Identify the dimensions
- Length \(L = 6\) meters
- Breadth \(B = 8\) meters
- Height \(H = 20\) meters

### Step 2: Substitute the dimensions into the formula
Now we substitute these values into the diagonal formula:

\[ D = \sqrt{6^2 + 8^2 + 20^2} \]

### Step 3: Calculate the squares
Calculate each square:
- \(6^2 = 36\)
- \(8^2 = 64\)
- \(20^2 = 400\)

### Step 4: Add the squares
Now add these values together:

\[ D = \sqrt{36 + 64 + 400} \]

\[ D = \sqrt{500} \]

### Step 5: Simplify the square root
We can simplify \(\sqrt{500}\):
\[ \sqrt{500} = \sqrt{100 \times 5} = \sqrt{100} \times \sqrt{5} = 10\sqrt{5} \]

### Conclusion
Thus, the length of the longest rod that can be placed in the room is:

\[ \boxed{10\sqrt{5}} \text{ meters} \]

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