Turning a corner with a pole
What is the length of the longest pole that can be carried horizontally around a corner at which a corridor that is a ft wide and a corridor that is b ft wide meet at right angles?

Verified step by step guidance

1. Visualize the problem by drawing a diagram of the two corridors meeting at a right angle, labeling the widths of the corridors as 'a' and 'b'.
2. Identify the relationship between the length of the pole and the geometry of the corridors, noting that the pole must fit within the confines of the two corridors as it turns the corner.
3. Use the Pythagorean theorem to express the length of the pole in terms of the distances from the corner to the points where the pole touches the walls of the corridors.
4. Set up an equation that represents the length of the pole as a function of the angle at which it is held while turning the corner.
5. Differentiate the function to find the maximum length of the pole, and solve for the critical points to determine the longest possible length.