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Yanka [14]
3 years ago
10

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 ≤

Mathematics
1 answer:
dusya [7]3 years ago
7 0
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.
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3 years ago
What is the approximate diameter of a sphere with a volume of 34 cm
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Answer:

<h3>The diameter is 4cm</h3>

Step-by-step explanation:

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To find the diameter we must first find the radius

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That's

34 =  \frac{4}{3} \pi {r}^{3}  \\  \\ 34 \times 3 = 4\pi {r}^{3}  \\  \\ 102 = 4\pi {r}^{3}  \\  {r}^{3}   = \frac{102}{4} \pi \\  \\ r =  \sqrt[3]{ \frac{51}{2\pi}}  \\  \\ r = 2.01 \\  \\ r = 2.0 cm

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6 0
3 years ago
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1. Number of boys in A is 52028.

2. Number of girls in B is 46510.

3. Town B has more number of girls.

Step-by-step explanation:

1. 95364 - 43336 = 52028

2. 100005-53495=46510

3. 43336 (A < B)

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3 years ago
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