[IMG]http://i115.photobucket.com/albums/n297/pakman134/optimizationsled.jpg[/IMG]
This really belongs in the "advance applied math section"
For question 1. you should know that the maximum value that a friction force is where R is the normal reaction force between the object and the surface it in in contact with and is the coefficient of friction.
(if you don't follow what I say here ask me to post a diagram)
the normal reaction force is given by . therefore the maximum friction force is . If there is to be motion the forward must be greater than the maximum friction force (such that there is a net resultant force forwards).
so we get
which rearranges to
this can rearrange to give
to find the max/min or F you differentiate using the chain rule. to get
we require
which rearranges to
using some simple trig you get and
put this into the formula (with your value of ) I get the minimum value of .
you should be able to notice that F will be a maximum when in which case
Edit: F is actually maximum in the case when in this case you would just be pulling the sledge vertically
I somewhat follow what you're saying and when I follow your process I come up with the same answers as you did, but with your answers they're still coming up as wrong. Maybe it is a simple math error but I can't seem to see anything really wrong with your process. Hm.....