Room height from two diagonals: The longest rod that can lie flat on the floor of a rectangular room is 10 m (the floor diagonal). The longest rod that can fit inside the room is 10√2 m (space diagonal). Find the height of the room (in m).

Introduction / Context:This is a 3D Pythagoras application. The floor diagonal gives √(l^2 + b^2); the space diagonal gives √(l^2 + b^2 + h^2). Using both, we can isolate the height h of the room.

Given Data / Assumptions:

• Floor diagonal = 10 m ⇒ √(l^2 + b^2) = 10
• Space diagonal = 10√2 m ⇒ √(l^2 + b^2 + h^2) = 10√2

Concept / Approach:

• Use the identity: (space diagonal)^2 − (floor diagonal)^2 = h^2.

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

• 8 m, 7.5 m, 6 m: Each gives space diagonal less than 10√2, contradicting the given value.

Common Pitfalls:

• Confusing the floor diagonal with a wall diagonal.
• Not squaring correctly when applying Pythagoras in 3D.

Final Answer:

10 m