To solve this problem of calculating the diagonal of the rectangular solid (parallelepiped), we use the 3D Pythagorean theorem. The diagonal $d$ of a rectangular solid with dimensions $l$, $w$, and $h$ is given by:

$d = \sqrt{l^2 + w^2 + h^2}$

Here, the given dimensions are:

• $l = 12 \, \text{cm}$,
• $w = 7 \, \text{cm}$,
• $h = 5 \, \text{cm}$.

Step 1: Plug the values into the formula

$d = \sqrt{12^2 + 7^2 + 5^2}$

Step 2: Square the values

$d = \sqrt{144 + 49 + 25}$

Step 3: Add the squared values

$d = \sqrt{218}$

Step 4: Take the square root

$d \approx 14.76 \, \text{cm}$

Final Answer:

The diagonal of the parallelepiped is approximately 14.76 cm.

Would you like a detailed explanation of the formula or further assistance? Here are some related questions:

1. How does the Pythagorean theorem extend to 3D space?
2. What are other real-life applications of calculating the diagonal of a rectangular prism?
3. How can we generalize this formula for higher dimensions?
4. What tools can be used to calculate the square root in complex cases?
5. Can the formula be derived using vector math?

Tip: Always double-check units and ensure consistency when working with dimensions in practical applications.