First of all, I wonder: in $F=ma$ does the acceleration have to be constant? I believe so but, just as confirmation.
A 4.80 kg bucket of water is accelerated upward by a cord of negligible mass whose breaking strength is 75 N. If the bucket starts from rest, what is the minimum time required to raise the bucket a vertical distance of 12 m without breaking the cord?
I just put in 75N, and and the mass into $f=ma$, and I get the maximum acceleration, which was wrong, so I realized that the bucket itself must exert a force onto the rope.
Putting the mass of the bucket into $ma$, and the gravitational acceleration I got 47.088 newton. 75-47.088=27.912. The rope can only have a force dragging it which is the maximum 27.912 newton before it breaks. Putting this into the formula again $f=ma$, $a=f/m$ to find the maximum acceleration, before the rope breaks. Then I put this into the distance formula. $$ x=x_0+v_0t+\frac12at^2, $$ to solve for the time which was 2.03 seconds.
Was this right? I have to ask for confirmation on this as studying alone in the summer is hard. Often there are only small questions we need answers to, to learn something big.