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.  Theta 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.<o:p></o:p>

sin theta = A/x     cos theta = B/y<o:p></o:p>

x = A/sin theta      y = B/cos theta<o:p></o:p>

L = x + y = A/sin theta  +  B/cos theta<o:p></o:p>

To find the minimum length (or the max allowable), differentiate the above equation to get:<o:p></o:p>

 - Acos theta/(sin theta)squared + Bsin theta/(cos theta)squared = 0<o:p></o:p>

or   Acos theta/(sin theta)squared = Bsin theta/(cos theta)squared<o:p></o:p>

  rewrite to   A/B = (tan theta)cubed  or  tan theta = cube root of A / cube root of B<o:p></o:p>

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.<o:p></o:p>

Sin theta = (cube root of A) / H      cos theta = (cube root of B) / H<o:p></o:p>

L = A/sin theta + B/cos theta  = A/(cube root of A) / H + B/(cube root of B) / H<o:p></o:p>

Simplifying, min. L = (A to the 2/3 power plus B to the 2/3 power) raised to the 3/2 power.<o:p></o:p>