Finding the Width of a Corridor

Date: 11/13/95 at 19:42:41
From: Anonymous
Subject: The differential, extremas

A steel girder 27 ft long is moved horizontally along a passageway
8 ft wide and into a corridor at right angles to the passageway.
How wide must the corridor be for the girder to go around the corner?
Neglect the horizontal width of the girder.

Date: 11/15/95 at 16:20:36
From: Doctor Ken
Subject: Re: The differential, extremas

Hello!

As your subject line indicates, this is a problem that will use Calculus
and local min/max stuff.  Here's a hint on how you might set it up:

                     |              |
                     |              |
                     |            / |
                     |          /   |
                     |        /     |
                     |      /       |
                     |    /         |
                     |  /           |
_____________________|/             |  _
                    / 27 ft.        |  |
                  /                 |  |
                /                   | 8 ft.
              /                     |  |
            /                       |  |
__________/_________________________|  -
                     |---- x ft. ---|

So you want to find the minimum value of x such that the girder can 
still go around the corner, right?  Well, for any value of x, there will 
be a "sticking point" where the girder will fit most tightly, i.e. where 
the girder's ends are closest to the walls of the corridors as the 
girder travels around the bend.  So we want to find the value of x where 
this sticking point makes both ends of the ladder exactly touch the 
walls.  See if you can find some similar triangles to help you out.

From here I'll direct you to a sketch I made using "The Geometer's 
Sketchpad."  If you have access to a Macintosh, you can look at the 
sketch by downloading a demo of the program from the URL 
http://mathforum.org/dynamic.html   , and then you can download 
the sketch I made by using the URL 

http://mathforum.org/dr.math/sketches/girder.gsp   

Enjoy!

-Doctor Ken,  The Geometry Forum