the diagonal is the longest distance in a cuboid (the shape of the room is a cuboid) hence, length of the longest pole that can be put in the room= room's diagonal = cuboids space diagonal with dimensions a = 10m, b = 10m and c = 5m

using the Pythagorean theorem we find the length of the space diagonal

d = (a^2 + b^2 + c^2)^(1/2)

by replacing and simplifying we find

d = 15m

to see the step by step solution click here:

Click to see all the steps

the longest pole to fit the room will be 15 meters long.