The problem asks to find the largest possible volume of a box with integer dimensions, where the diagonal $XY$ is 9. The box has rectangular faces, meaning its sides are perpendicular, and the long diagonal can be calculated using the Pythagorean Theorem in 3D.

Step-by-step solution:

1. Diagonal Formula for a Rectangular Box: The diagonal $XY$ of the box with dimensions $a$, $b$, and $c$ is given by: $XY = \sqrt{a^2 + b^2 + c^2}$ We know that $XY = 9$, so: $\sqrt{a^2 + b^2 + c^2} = 9$ Squaring both sides: $a^2 + b^2 + c^2 = 81$

2. Maximizing the Volume: The volume $V$ of the box is: $V = a \times b \times c$ We need to find integer values of $a$, $b$, and $c$ that satisfy $a^2 + b^2 + c^2 = 81$, while maximizing $V$.

3. Finding Integer Solutions: We now search for integer solutions to the equation $a^2 + b^2 + c^2 = 81$.

   Possible values for $a^2$, $b^2$, and $c^2$ are perfect squares less than or equal to 81: $0, 1, 4, 9, 16, 25, 36, 49, 64$.

   Let's try different combinations:

   • $a = 4$, $b = 4$, $c = 7$: $a^2 + b^2 + c^2 = 16 + 16 + 49 = 81$ So, $a = 4$, $b = 4$, and $c = 7$ is a valid solution.

4. Calculate the Volume: For $a = 4$, $b = 4$, and $c = 7$, the volume is: $V = 4 \times 4 \times 7 = 112$

Thus, the largest possible volume is 112.

Answer: (E) 112

Would you like more details on any part of the explanation, or do you have any questions?

Here are some related questions you might find useful:

1. How is the diagonal of a rectangular box derived?
2. What other integer solutions could exist for the equation $a^2 + b^2 + c^2 = 81$?
3. What is the importance of maximizing volume in optimization problems?
4. How do different combinations of dimensions affect the shape of the box?
5. Could the diagonal formula be applied to non-rectangular objects?

Tip: Always consider the geometric properties of objects when working with 3D problems.