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denis23 [38]
3 years ago
12

Line l is parallel to line m. The slope of line l is 2/3. What is the slope of line m

Mathematics
1 answer:
DochEvi [55]3 years ago
5 0

Answer:

2/3

Step-by-step explanation:

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Use the Divergence Theorem to evaluate S F · dS, where F(x, y, z) = z2xi + y3 3 + sin z j + (x2z + y2)k and S is the top half of
GenaCL600 [577]

Close off the hemisphere S by attaching to it the disk D of radius 3 centered at the origin in the plane z=0. By the divergence theorem, we have

\displaystyle\iint_{S\cup D}\vec F(x,y,z)\cdot\mathrm d\vec S=\iiint_R\mathrm{div}\vec F(x,y,z)\,\mathrm dV

where R is the interior of the joined surfaces S\cup D.

Compute the divergence of \vec F:

\mathrm{div}\vec F(x,y,z)=\dfrac{\partial(xz^2)}{\partial x}+\dfrac{\partial\left(\frac{y^3}3+\sin z\right)}{\partial y}+\dfrac{\partial(x^2z+y^2)}{\partial k}=z^2+y^2+x^2

Compute the integral of the divergence over R. Easily done by converting to cylindrical or spherical coordinates. I'll do the latter:

\begin{cases}x(\rho,\theta,\varphi)=\rho\cos\theta\sin\varphi\\y(\rho,\theta,\varphi)=\rho\sin\theta\sin\varphi\\z(\rho,\theta,\varphi)=\rho\cos\varphi\end{cases}\implies\begin{cases}x^2+y^2+z^2=\rho^2\\\mathrm dV=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi\end{cases}

So the volume integral is

\displaystyle\iiint_Rx^2+y^2+z^2\,\mathrm dV=\int_0^{\pi/2}\int_0^{2\pi}\int_0^3\rho^4\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi=\frac{486\pi}5

From this we need to subtract the contribution of

\displaystyle\iint_D\vec F(x,y,z)\cdot\mathrm d\vec S

that is, the integral of \vec F over the disk, oriented downward. Since z=0 in D, we have

\vec F(x,y,0)=\dfrac{y^3}3\,\vec\jmath+y^2\,\vec k

Parameterize D by

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

where 0\le u\le 3 and 0\le v\le2\pi. Take the normal vector to be

\dfrac{\partial\vec r}{\partial v}\times\dfrac{\partial\vec r}{\partial u}=-u\,\vec k

Then taking the dot product of \vec F with the normal vector gives

\vec F(x(u,v),y(u,v),0)\cdot(-u\,\vec k)=-y(u,v)^2u=-u^3\sin^2v

So the contribution of integrating \vec F over D is

\displaystyle\int_0^{2\pi}\int_0^3-u^3\sin^2v\,\mathrm du\,\mathrm dv=-\frac{81\pi}4

and the value of the integral we want is

(integral of divergence of <em>F</em>) - (integral over <em>D</em>) = integral over <em>S</em>

==>  486π/5 - (-81π/4) = 2349π/20

5 0
3 years ago
Factor completly x^2 -64
Hoochie [10]

∑ Hey, jillianwagler ⊃

Answer:

\left(x+8\right)\left(x-8\right)

Step-by-step explanation:

<u><em>⇒ Given info:</em></u>

<em>Factor completely </em>x^2 -64

<u><em>⇒ Solution:</em></u>

<em>Rewrite 64 as 8 ² because 8² is equal to 64. 8 ² = 8 × 8 = 64.</em>

<em />=x^2-8^2

Applying difference of two squares formula: x^2-y^2=\left(x+y\right)\left(x-y\right)

=x^2-8^2=\left(x+8\right)\left(x-8\right)\\

=\left(x+8\right)\left(x-8\right)

Answer~: \left(x+8\right)\left(x-8\right)

<u><em>xcookiex12</em></u>

<em>8/19/2022</em>

7 0
2 years ago
Amanda and her four sisters divided 1,021 stickers equally how many did each girl receive
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1021 / 5 = 204.20.....each girl got 204 stickers.....with a remainder of 1 sticker
7 0
3 years ago
In winter, the price of apples suddenly went up by 0.75 per pound. Sam bought 3 pounds of apples at the new price for a total of
forsale [732]
5.88 / 3 = 1.96 per pound
original price - 1.21 per pound
5 0
3 years ago
Find the inverse of the function y = 2x2 + 2
Anvisha [2.4K]
We have that
y = 2x²<span> + 2

step 1

</span><span>exchange the value of x for y and the value of y for x
</span>y = 2x² + 2------> x=2y²+2

step 2
clear y variable
 x=2y²+2----> 2y²=x-2----> y²=[(x-2)/2]-----> y=(+/-)√[(x-2)/2]

step 3
the inverse is
f(x)-1=(+/-)√[(x-2)/2]

8 0
3 years ago
Read 2 more answers
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