Let divf = 6(5 − x) and 0 ≤ a, b, c ≤ 12. (a) find the flux of f out of the rectangular solid 0 ≤ x ≤ a, 0 ≤ y ≤ b, and 0 ≤ z ≤ c. your answer will be in terms of a, b, c. flux = (b) for what values of a, b, c is the flux the largest? what is that largest flux?

Continuing from the setup in the question linked above (and using the same symbols/variables), we have






The next part of the question asks to maximize this result - our target function which we'll call - subject to .

We can see that is quadratic in , so let's complete the square.



Since are non-negative, it stands to reason that the total product will be maximized if vanishes because is a parabola with its vertex (a maximum) at (5, 25). Setting , it's clear that the maximum of will then be attained when are largest, so the largest flux will be attained at , which gives a flux of 10,800.

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