In a rectangular room measuring 12 m in length, 8 m in breadth and 9 m in height, what is the length of the longest pole that can be placed inside the room from one corner to the opposite corner?

Introduction / Context:
This problem is a classic application of three-dimensional Pythagoras theorem. You are asked to find the longest pole that can fit inside a rectangular room. That longest pole will lie along the space diagonal of the cuboid formed by the room, connecting one vertex to the opposite vertex. The question tests your understanding of distances in 3D geometry and how to extend the Pythagorean theorem from two to three dimensions.

Given Data / Assumptions:

• Length of the room, l = 12 m.
• Breadth (width) of the room, b = 8 m.
• Height of the room, h = 9 m.
• The room is rectangular in all three dimensions.
• The longest pole lies along the space diagonal of this rectangular box.

Concept / Approach:
In a cuboid (rectangular box) with sides l, b and h, the length of the space diagonal d is given by the formula:
d = sqrt(l^2 + b^2 + h^2) This comes from applying Pythagoras theorem twice: first to find the diagonal of the base rectangle, then to combine that with the height. Once we compute d using the given dimensions, we choose the matching option in metres.

Step-by-Step Solution:
Compute l^2 = 12^2 = 144. Compute b^2 = 8^2 = 64. Compute h^2 = 9^2 = 81. Sum of squares = 144 + 64 + 81 = 289. Space diagonal d = sqrt(289). Since 289 = 17^2, d = 17 m.

Verification / Alternative check:
We can cross-check by remembering that 17 is part of the Pythagorean triple 8–15–17. Here the numbers are different, but the square sum 289 exactly equals 17^2, so there is no rounding. Also, among the options, 17 m is the only value consistent with the exact square root of 289. Therefore, the calculation is reliable and the pole length of 17 m is correct.

Why Other Options Are Wrong:
Values 14 m, 15 m, and 16 m all correspond to smaller squared lengths: 196, 225 and 256 respectively, which do not match 289. A value like 18 m would give a squared length of 324, which is greater than 289 and does not describe the space diagonal for these dimensions. Hence none of these other options fits the formula for the diagonal of this room.

Common Pitfalls:
A common error is to compute only the diagonal of the floor (using 12 m and 8 m) and forget to include the height, which gives a shorter length than the true space diagonal. Another mistake is mixing up units, but here everything is in metres, so consistent usage avoids that issue. Students should also remember to square and add correctly before taking the square root.

Final Answer:
The length of the longest pole that can be placed inside the room is 17 m.