Given some water level ~X~, it is easy to compute the total volume of water ~V(X)~. Basically, one has to compute for each object whether it is floating or touching the lid. Luckily, ~V(X)~ is monotonic in ~X~. We need to find such ~X~ that ~V(X) = P~, where ~P~ is the volume of water in the input data. This can be done using binary search.

Another solution, which is much more educating, is to observe that the set of possible water levels can be split into a disjoint set of intervals such that within every interval each object is either floating or touches the lid. This allows solving the task using a linear equation. This is the so-called "scanning line" principle. Unfortunately, we do not see any way to force the contestants to use this method.