The problem asks to find the length of diagonal $\overline{AC}$ in the rectangular solid. We are provided with the dimensions of the solid, where $AB = 5 \, \text{m}$ and $AD = 2 \, \text{m}$, both measured along the edges of the solid.

Steps to Solve

To find the length of the diagonal $\overline{AC}$, we need to apply the Pythagorean theorem in 3D.

1. First, find the diagonal $\overline{AC}$ in the base of the solid, which is a rectangle formed by $A, B, C, D$. The length of diagonal $AC$ in the base can be found using the Pythagorean theorem in the rectangle:

   $AC^2 = AB^2 + AD^2$

   Using the given values:

   $AB = 5 \, \text{m}, \quad AD = 2 \, \text{m}$

   Substituting:

   $AC^2 = 5^2 + 2^2 = 25 + 4 = 29$

   So,

   $AC = \sqrt{29} \, \text{m} \approx 5.39 \, \text{m}$

Therefore, the length of the diagonal $\overline{AC}$ is approximately 5.39 meters.

Would you like further details or have any questions about this solution?

Here are 5 related questions to deepen understanding:

1. How would you find the diagonal of a cube using the same method?
2. What is the formula for the diagonal in a 3D rectangular prism in general?
3. If the height of the rectangular prism was given, how would you calculate the space diagonal?
4. What are the units for the diagonal if the dimensions were in centimeters instead of meters?
5. How does the Pythagorean theorem extend from 2D to 3D geometry?

Tip: Always check if you need to apply the Pythagorean theorem in 2D or 3D depending on the problem context.