# Length of stick by derivation

1. Dec 15, 2011

### georg gill

one want to find the length of the two hypotenuses of the figure

Here is what I did. I looked at the small first with hypotenus unknown and adjacent=1 and the other cathetus unknown. Then i wanted to find max value of the smallest hypotenus.

$$tan\theta=\frac{1}{a}$$

$$a=\frac{1}{tan\theta}$$ Smallest hypotenus=sh

$$sh^2=1+\frac{1}{tan^2\theta}$$

i took derivative to find max value

$$sh=\sqrt{1+cot^2\theta}$$

$$\frac{d}{dx}sh=\frac{1}{\sqrt{1+cot^2\theta}} \cdot \frac{coth\theta}{sin^2\theta}$$

but this is equal to zero when $$\theta=\frac{\pi}{2}$$ whoch gives sh=1

which is obviously wrong since adjacent cathetus=1

What is wrong?

Last edited: Dec 15, 2011
2. Dec 15, 2011

### mathman

Your question is confusing. Do you want the stick length for a given θ or something else? It is not clear at all.

3. Dec 15, 2011

### georg gill

sorry. One want to find how long the stick could be and still fit in in the hallway with height 1 and chamber with width 8. So another way to try to describe problem, how long the stick could be and still fit in the drawing

4. Dec 16, 2011

### mathman

If there is no limit on the vertical (in the picture) side of the chamber, you can place the stick vertically and it will be infinite in length. There must be some further constraint in the problem.

5. Dec 16, 2011

### georg gill

I am not sure. It says it is being carried horizontally in the assignment text. This is word by word translation:

You are going to carry a stick from a one meter high hallway into a chamber that has eight meters in width. The stick is carried horizontally and we assume that its thickness is infinite small. How long could the stick be maximally?

6. Dec 17, 2011

### JHamm

This might be hard to describe online but I'll give it a go. I'd set up a function to give the height of the ladder as a function of the height of the bottom of the ladder, set the zero point as the length of the ladder minus 1m from the top of the hallway. Set it up so that the bottom of the ladder sits against the right wall and intersects the corner. Differentiate the function and you can find the maximum height above this zero point, solve this for the max height being the length of the ladder.
Alternatively make the function dependent on the angle theta if that seems simpler to you.

7. Dec 17, 2011

### mathman

Is this the question? What is the longest possible stick that can be moved from the chamber to the hall?