I had an HP 870cxi that had the same problem. You can try washing the rollers with a damp rag and see if the problem goes away. In my case it was the rollers - they would get dirty and then the paper would slip during the feed process. Really frustrating, as the printer overall was quite good.
Your problems are essentially one and the same apart from some minor details. Obviously, with the first problem you could solve the equation for the ellipse for one variable, say z. Then plug this into the equation for the volume V = xyz, take the first partial derivatives with respect to x...
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