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2 years ago

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The area of a square is s2, where s is the side length. Suppose you have three squares. One square has side length 9 feet. Anot

her square has side length 6 feet. The third square has side length . Is the sum of the areas of the two smaller squares equal to the area of the large square? Use pencil and paper. Explain your answer. Then describe three squares for which you would give the opposite answer for the same question.

1 answer:

dlinn [17]2 years ago

6 0

Answer:

Dead Trollz

Step-by-step explanation:

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Problem 4: Let F = (2z + 2)k be the flow field. Answer the following to verify the divergence theorem: a) Use definition to find

Viktor [21]

Given that you mention the divergence theorem, and that part (b) is asking you to find the downward flux through the disk $x^2+y^2\le3$ , I think it's same to assume that the hemisphere referred to in part (a) is the upper half of the sphere $x^2+y^2+z^2=3$ .

a. Let $C$ denote the hemispherical <u>c</u>ap $z=\sqrt{3-x^2-y^2}$ , parameterized by

$\vec r(u,v)=\sqrt3\cos u\sin v\,\vec\imath+\sqrt3\sin u\sin v\,\vec\jmath+\sqrt3\cos v\,\vec k$

with $0\le u\le2\pi$ and $0\le v\le\frac\pi2$ . Take the normal vector to $C$ to be

$\vec r_v\times\vec r_u=3\cos u\sin^2v\,\vec\imath+3\sin u\sin^2v\,\vec\jmath+3\sin v\cos v\,\vec k$

Then the upward flux of $\vec F=(2z+2)\,\vec k$ through $C$ is

$\displaystyle\iint_C\vec F\cdot\mathrm d\vec S=\int_0^{2\pi}\int_0^{\pi/2}((2\sqrt3\cos v+2)\,\vec k)\cdot(\vec r_v\times\vec r_u)\,\mathrm dv\,\mathrm du$

$\displaystyle=3\int_0^{2\pi}\int_0^{\pi/2}\sin2v(\sqrt3\cos v+1)\,\mathrm dv\,\mathrm du$

$=\boxed{2(3+2\sqrt3)\pi}$

b. Let $D$ be the disk that closes off the hemisphere $C$ , parameterized by

$\vec s(u,v)=u\cos v\,\vec\imath+u\sin v\,\vec\jmath$

with $0\le u\le\sqrt3$ and $0\le v\le2\pi$ . Take the normal to $D$ to be

$\vec s_v\times\vec s_u=-u\,\vec k$

Then the downward flux of $\vec F$ through $D$ is

$\displaystyle\int_0^{2\pi}\int_0^{\sqrt3}(2\,\vec k)\cdot(\vec s_v\times\vec s_u)\,\mathrm du\,\mathrm dv=-2\int_0^{2\pi}\int_0^{\sqrt3}u\,\mathrm du\,\mathrm dv$

$=\boxed{-6\pi}$

c. The net flux is then $\boxed{4\sqrt3\pi}$ .

d. By the divergence theorem, the flux of $\vec F$ across the closed hemisphere $H$ with boundary $C\cup D$ is equal to the integral of $\mathrm{div}\vec F$ over its interior:

$\displaystyle\iint_{C\cup D}\vec F\cdot\mathrm d\vec S=\iiint_H\mathrm{div}\vec F\,\mathrm dV$

We have

$\mathrm{div}\vec F=\dfrac{\partial(2z+2)}{\partial z}=2$

so the volume integral is

$2\displaystyle\iiint_H\mathrm dV$

which is 2 times the volume of the hemisphere $H$ , so that the net flux is $\boxed{4\sqrt3\pi}$ . Just to confirm, we could compute the integral in spherical coordinates:

$\displaystyle2\int_0^{\pi/2}\int_0^{2\pi}\int_0^{\sqrt3}\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi=4\sqrt3\pi$

4 0

3 years ago

PLEAS HELP I REALLY NEED HELP

blsea [12.9K]

Answer:

the first one in the picture

Step-by-step explanation:

8x8x8x8=4096

9x9x9x9=6561

4096x6561=26873856

72x72x72x72=26873856

5 0

2 years ago

1 Select the correct answer. Which rate is equivalent to ? A. B. C. D.

ki77a [65]

D that is the answer have a good day D.

7 0

3 years ago

What is the y-intercept of the graph with equation f(x) = 4x^2 - 10x + 7 ?

Nutka1998 [239]

Answer:

The y intercept to this problem is;

(0,7)

Explanation:

To find the x-intercept, substitute in 0 for y and solve for x . To find the y-intercept, substitute in 0 for x and solve for y .

4 0

2 years ago

Simplify 6^-6/6^-5 . rewrite the expression in the form 6^n

HACTEHA [7]

Answer: $=\frac{1}{6}\quad \left(\mathrm{Decimal:\quad }\:0.16666\dots \right)$

Step-by-step explanation:

$\frac{6^{-6}}{6^{-5}}$

$=6^{-6-\left(-5\right)}$

$=6^{-6+5}$

$=6^{-1}$

$=\frac{1}{6}$

6 0

3 years ago