Word Problem: Max/ Min

[NYC]

OK, a person is carrying a block of wood down a corrider with width of 8 meters, and is attempting to turn into a corridor with width of 4 meters. What is the maximum length this wood can be?
I....have no clue as to how I can start this. In my diagrams, I wind up drawing an a quarter of an ellipse in a rectangle of sides 8 and 4. If you've got any suggestions, that would be great...I'm thinking that you might need to find the maximum area of a triangle formed by the wood, the ends of the corridors...and well, thats just speculation.

 

[soroban]

Hello, NYC!

A person is carrying a block of wood down a corrider with width of 8 meters,
and is attempting to turn into a corridor with width of 4 meters.
What is the maximum length this wood can be?

The word "block" is a very poor choice.
It should be a <u>ladder</u> or a board . . . and its thickness is considered negligible.

      *-------------------*---- 
      |                 / :
      |             L2/   :4
      |             / θ   :
      |           *------------       Let the ladder be comprised
      |         / |
      |       /   |                   of two segments: L1 and L2.
      |   L1/     |
      |   /       |
      | / θ       |
      * - - - - - |
      |     8     |

In the lower right triangle: .\(\displaystyle \cos\theta\,=\,\frac{8}{L_1}\;\;\Rightarrow\;\;L_1\,=\,8\cdot\sec\theta\)

In the upper right triangle: .\(\displaystyle \sin\theta\,=\,\frac{4}{L_2}\;\;\Rightarrow\;\;L_2\,=\,4\cdot\csc\theta\)

Hence, we have: .\(\displaystyle \L L\:=\:8\cdot\sec\theta\,+\,4\cdot\csc\theta\)

. . and that is the function we must maximize.

 

[NYC]

OK

Aye, thanks.
I did it another way, and got an answer of 16.647... which is the correct answer apparently
I would like to know a little more about your method though.
I graphed the function, and it appeared to be unusable (it looks to have several asymptotes).
My method included slope to get the variables in terms of each other, and then I plugged that into the distance formula.