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-->|

Let's call the portion of the panel from the inside corner to the wall of corridor A x, and the portion from the corner to the wall of B y.  Then x + y = Length.  ¥è is the angle formed between the panel and the side wall of A, and also between the panel and a line parallel to width B.

sin ¥è = A/x     cos¥è = B/y

x = A/sin¥è      y = B/cos¥è

L = x + y = A/sin¥è  +  B/cos¥è

To find the minimum length (or the max allowable), differentiate the above equation to get:

 - Acos¥è/sin©÷¥è + Bsin¥è/cos©÷¥è = 0

or   Acos¥è/sin©÷¥è = Bsin¥è/cos©÷¥è

  rewrite to   A/B = tan©ø¥è  or  tan¥è = ©ø¡îA / ©ø¡îB

So picture a right triangle with sides of cube root of A and cube root of B.  The hypotenuse H equals the square root of the sum of the cube root of A squared and the cube root of B squared.

sin¥è = ©ø¡îA / H      cos¥è = ©ø¡îB / H

L = A/sin¥è + B/cos¥è  = A/©ø¡îA / H + B/©ø¡îB / H

Simplifying, L = the sum of A to the 2/3 power and B to the 2/3 power, the sum raised to the 3/2 power.