The woman is covered in 3 different kinds of power series (all Taylor, but one is general and the other two are specific to 1/(1-x) and ln x, respectively) that certain kinds of scientists (presumably personified here by the man) love to swap in for more complicated terms by waving their hands and chanting “first order”. Truncated series give fugly equations a more tractable form by applying certain assumptions (e.g., x is very small, x is very large, or x is fuck-it-we-ball).
EDIT: nvm apparently this is a pop culture reference.
Roughly a truncated cone with diameters ~7 nuts and ~9 nuts, and the cup is ~12 nuts high (loose guesses, it is hard to tell due perspective and nuts of different sizes). Throw in an extra layer to account for the heap at the top (which is a dome taller than 1 hazelnut, but treating it as a shorter but full layer should give some error cancellation) to give a height of 13. The volume of a truncated cone of those dimensions is ~657 cubic hazelnut diameters. Random sphere packing is 64% space-efficient (though wall effects should decrease this number) giving a total of 420 nuts (nice).
Multiple edits for clarity and typos.
Answer
This ends up being about 5% lower than the true answer. I’m surprised it’s that close. This is in the opposite direction from what I expected given wall effects (which would decrease the real number relative to my estimate). Perturbing one of the base diameters by 1 nut causes a swing of ~50, so measurement error is quite important.