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Taya2010 [7]
2 years ago
15

4. Consider all possible rectangular solids defined by 0 ≤ x ≤ a, 0 ≤ y ≤ b, 0 ≤ z ≤ 1 and the vector field given by F = (−x 2 −

4xy)i − 6yzj + 12zk. (a) Calculate the total outwards flux through the 6 sides of the solid. (b) Find value(s) for a and b that yield the maximum flux. (c) Calculate the maximum total flux.
Mathematics
1 answer:
N76 [4]2 years ago
7 0

Step-by-step explanation:

By using the Divergence Theorem, the flux equals

∫∫∫ div F dV

= ∫∫∫ ((-2x - 4y) + (-6z) + 12) dV

= ∫(z = 0 to c) ∫(y = 0 to b) ∫(x = 0 to a) (12 - 2x - 4y - 6z) dz dy dx

= ∫(z = 0 to c) ∫(y = 0 to b) (12a - a^2 - 4ay - 6az) dy dx

= ∫(z = 0 to c) (12ab - a^2 b - 2ab^2 - 6abz) dx

= 12abc - a^2 bc - 2ab^2 c - 3abc^2.

So, we need to find the maximum value of

f(a,b,c) = 12abc - a^2 bc - 2ab^2 c - 3abc^2, with a,b,c > 0.

First, we find the critical points of f.

f_a = 12bc - 2abc - 2b^2 c - 3bc^2 = bc(12 - 2a - 2b - 3c)

f_b = 12ac - a^2 c - 4abc - 3ac^2 = ac(12 - a - 4b - 3c)

f_c = 12ab - a^2 b - 2ab^2 - 6abc = ab (12 - a - 2b - 6c).

Setting these equal to 0 (and remembering that a, b, c > 0):

12 - 2a - 2b - 3c = 0

12 - a - 4b - 3c = 0

12 - a - 2b - 6c = 0

Solving this system yields a = 3, b = 3/2, c = 1.

By the Second Derivative Test or otherwise, this can easily be checked to yield the maximum flux.

This maximal flux equals f(3, 3/2, 1) = 27/2

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Answer:

A. reflection across the y-axiss

Step-by-step explanation:

Given:

The locations of the two points are (-4 , 8) and (-4 , -8).

To find:

The relation between two points.

From the given points (-4, 8) and (-4 , -8), it is clear that the y-coordinates are same but the sign of x-coordinates are opposite.

If a figure is reflected across the y-axis, then we change the sign of x-coordinate and the y-coordinates remain same, i.e.,

(x,y)→(-x,y)

For (-4,8)

(-4,8)→ (-4,-8)

So, it is reflection across the y-axis.

Therefore, the correct option is A.

6 0
3 years ago
Angles R and S are complementary. If angle R is = 4x - 20 and angle S = 6x - 7, then angle R measures ___ degrees.
Shalnov [3]

Answer:

26.8°

Step-by-step explanation:

4x - 20 + 6x - 7 = 90

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10x= 117

x = 11.7

so, the angle R = 4x-20 =4(11.7)-20

= 46.8-20 = 26.8°

6 0
2 years ago
What is the answer? 15 points
artcher [175]

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8 0
1 year ago
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OPTION C

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4 0
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svp [43]

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42

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I hope this helps.

6 0
3 years ago
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