I'm not sure what happened on the last step: You have , which should give you:
, so x is approximately 0.7375 radians or 42.3 degrees.
That gives you a ladder length of .
Sure, but you already did the hard part...
Starting with your equation (recall you took the derivative and set it equal to zero):
add to both sides:
multiplly both sides by and combine the trig functions:
Then take the cube root and inverse tangent to get x=0.7375. But you want the length of the ladder, so you have to plug in to the original equation:
Interesting, I was just doing this problem in a book I'm reading. The solution I found doesn't use the angle at all, but instead labels the lengths with new variable names and finds a relationship between them in terms of the known variables. The answer is given in terms of and and I thought you might find it interesting. It's attached to my reply.