The internal dimensions of a rectangular box are `12 cm xx x cm xx 9 cm`. If the length of longest rod that can be placed in this box is 17 cm, find x.

Answer

To solve the problem, we need to find the value of \( x \) in the dimensions of a rectangular box (cuboid) given that the longest rod that can fit inside it is 17 cm. The internal dimensions of the box are \( 12 \, \text{cm} \times x \, \text{cm} \times 9 \, \text{cm} \).

### Step-by-Step Solution:

1. **Understand the Problem**:
We know that the longest rod that can fit inside a cuboid is equal to the length of the diagonal of the cuboid. The formula for the diagonal \( d \) of a cuboid with dimensions \( l \), \( b \), and \( h \) is given by:
\[ d = \sqrt{l^2 + b^2 + h^2} \]

2. **Set Up the Equation**:
Given the dimensions of the box as \( l = 12 \, \text{cm} \), \( b = x \, \text{cm} \), and \( h = 9 \, \text{cm} \), we can substitute these values into the diagonal formula:
\[ 17 = \sqrt{12^2 + x^2 + 9^2} \]

3. **Square Both Sides**:
To eliminate the square root, we square both sides of the equation:
\[ 17^2 = 12^2 + x^2 + 9^2 \]
This simplifies to:
\[ 289 = 144 + x^2 + 81 \]

4. **Combine Like Terms**:
Now, combine the constant terms on the right side:
\[ 289 = 225 + x^2 \]

5. **Isolate \( x^2 \)**:
Subtract 225 from both sides to isolate \( x^2 \):
\[ x^2 = 289 - 225 \]
\[ x^2 = 64 \]

6. **Take the Square Root**:
Finally, take the square root of both sides to find \( x \):
\[ x = \sqrt{64} \]
\[ x = 8 \, \text{cm} \]

### Conclusion:
The value of \( x \) is \( 8 \, \text{cm} \).