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

Answer

To find the value of \( x \) in the dimensions of the rectangular box, we can follow these steps:

### Step 1: Understand the problem
We have a rectangular box with internal dimensions \( 12 \, \text{cm} \times x \, \text{cm} \times 9 \, \text{cm} \). We need to find the value of \( x \) given that the longest rod that can fit inside the box is \( 17 \, \text{cm} \).

### Step 2: Use the formula for the diagonal of a rectangular box
The length of the longest rod that can fit inside the box is equal to the length of the diagonal of the box. The formula for the diagonal \( d \) of a rectangular box with dimensions \( l \), \( b \), and \( h \) is given by:
\[ d = \sqrt{l^2 + b^2 + h^2} \]
In our case, \( l = 12 \, \text{cm} \), \( b = x \, \text{cm} \), and \( h = 9 \, \text{cm} \).

### Step 3: Set up the equation
Since we know that the diagonal \( d = 17 \, \text{cm} \), we can set up the equation:
\[ 17 = \sqrt{12^2 + x^2 + 9^2} \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides gives us:
\[ 17^2 = 12^2 + x^2 + 9^2 \]
Calculating the squares:
\[ 289 = 144 + x^2 + 81 \]

### Step 5: Combine like terms
Now we combine the constants on the right side:
\[ 289 = 144 + 81 + x^2 \]
\[ 289 = 225 + x^2 \]

### Step 6: Isolate \( x^2 \)
To isolate \( x^2 \), subtract \( 225 \) from both sides:
\[ 289 - 225 = x^2 \]
\[ 64 = x^2 \]

### Step 7: Solve for \( x \)
Now, take the square root of both sides to find \( x \):
\[ x = \sqrt{64} \]
\[ x = 8 \]

### Final Answer
Thus, the value of \( x \) is \( 8 \, \text{cm} \).

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