Question

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?

Answer 1

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

$\displaystyle\iint_{\mathcal S}\mathbf f\cdot\mathrm d\mathbf S=\iiint_{\mathcal R}(\nabla\cdot f)\,\mathrm dV$
$=\displaystyle6\int_{z=0}^{z=c}\int_{y=0}^{y=b}\int_{x=0}^{x=a}(5-x)\,\mathrm dx\,\mathrm dy\,\mathrm dz$
$=\displaystyle6bc\int_0^a(5-x)\,\mathrm dx$
$=6bc\left(5a-\dfrac{a^2}2\right)=3abc(10-a)$

The next part of the question asks to maximize this result - our target function which we'll call $g(a,b,c)=3abc(10-a)$ - subject to $0\le a,b,c\le12$.

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

$g(a,b,c)=-3bc(a^2-10a+25-25)=3bc(25-(a-5)^2)$

Since $b,c$ are non-negative, it stands to reason that the total product will be maximized if $a-5$ vanishes because $25-(a-5)^2$ is a parabola with its vertex (a maximum) at (5, 25). Setting $a=5$, it's clear that the maximum of $g$ will then be attained when $b,c$ are largest, so the largest flux will be attained at $(a,b,c)=(5,12,12)$, which gives a flux of 10,800.

Answer 2

The largest flux will be 10,800.flux of f out of the rectangular solid 0 ≤ x ≤ a, 0 ≤ y ≤ b, and 0 ≤ z ≤c

What is flux?

Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance.

Flux is a concept in applied mathematics and vector calculus that has many applications to physics. For transport phenomena,

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

$\int\int_sf.ds=\iny\int\int_R(\Delta.f)dV$

             $=6\int\limits^c_0 \int\limits^b_0 \int\limits^a_0(5-x)dxdydz$

             $=6bc\int\limits^a_0 ({5-x}) \, dx$

              $=6bc (5a-\dfrac{a^2}{2})$

              $=3abc \ (10-a)$

The next part of the question asks to maximize this result - our target function which we'll call   $g(a,b,c)=3abc(10-a)$  subject to $0\leq a,b,c\leq12$

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

$g(a,b,c)=-3bc(a^2-10a+25-25)=3bc(25-(a-5)^2)$

Since bc are non-negative, it stands to reason that the total product will be maximized if a-5  vanishes because  25-(a-5)^2 is a parabola with its vertex (a maximum) at (5, 25). The Setting, a=5

it's clear that the maximum of g will then be attained when bc is largest, so the largest flux will be attained at (a,b,c)=(5,12,12) , which gives a flux of 10,800.

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