When my blanket is hanging off the bed, it's difficult to pull it up again. The more that hangs off, the harder it is; as we learn in calculus, pulling up the first half of the way is three times harder than pulling up the second half of the way if the part hanging off is rectangular.
I would like to find a shape that minimizes the average work done in pulling up the blanket, given that the blanket only hangs off one side of the bed (I.e. doesn't fold around the corner). So, given a random line cutting off part of the blanket, I'd like to minimize the work done in pulling that cut-off part back over the line.
However, it must at least cover my bed, which is queen-size.
Any suggestions?
Edit: the question as written is garbage. Consider thi variant: fix the area of a plane figure at 1. Given a random line that separates the figure, look at the work done in moving the smaller half across the line, assuming a force equal to gravity opposing the move. What shape minimizes average work done?
Thinking by what I understand, I think that you are going towards a size of rectangular size of blanket with less dimensions such that you can take it and it does not 'hang' from one side. See the reason a blanket is a "uniform" quadrilateral is because it could cover you from every side. If your blanket is a parallelogram, (which according to me is hilarious,) it might not cover you from the side where it is aligned. So the "perfect" shape for a blanket is a rectangle.
You might use another oversized blanket with another shape, then you might go for a circle, since it's unique shape offers minimum area when its hanging from your bed(unless it is half of the blanket, particularly the diameter) but it might not cover you from every side while you are sleeping. But you can always tuck your sheet in so that it doesn't hang.
And here I only discussed the maths part of what I saw in this question and i do not think it is "actually" a maths question since you are only talking about shapes and the probability that a side of blanket would fall or not. You cannot alter that probability basically by it's shape. It's shape should be suited so as to cover you from every side. Basically, just ask the shopkeeper for a blanket of lesser dimensions.