A large panel needs to be moved through a corridor, the panel is tall as the corridor. The corridor is A feet wide before a right angle turn, after the turn, it is B feet wide. What is the maximum length of the panel that can pass through this corner.

Overhead view of the hallway:

+------------+---
|           /   |
|          /    |B ft
|         /     |
|        /+------
|       / |
|      /  |
|     /   |
|    /    |
|   /     |
|  /      |
| /       |
|/        |
+<-A ft-->|

I'll rewrite this now.  The formatting buttons seem to have really gone haywire.

I decided to simplify things by scaling A and B by dividing A.

This creates new widths of 1 and B/A.  Let R=B/A

My steps are much the same.  The function to be minimized 

Sqrt(x^2 + y^2) becomes (x/(x-1))Sqrt((x-1)^2 - R^2)

[I wish I had thought to ignore the square root like Fredeico did.]

The derivative eventually reduces to (x-1)^3 -R^2 and is equal to zero when x= R^(2/3)+1

Sqrt(x^2 + y^2) becomes (R^(2/3) + 1)^(3/2)

This is equal to 2Sqrt(2) when R=1

In terms of the original A and B the solution is:

A((B/A)^(2/3)+1)^(3/2)

The angle at the tightest fit is Arctan((R+R^(1/3))/(R^(2/3)+1)) which is only 45degrees when R=1

-Jer