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3 years ago

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 F) - (integral over D) = integral over S

==> 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:

⇒ Given info:

Factor completely $x^2 -64$

⇒ Solution:

Rewrite 64 as 8 ² because 8² is equal to 64. 8 ² = 8 × 8 = 64.

$=x^2-8^2$

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

xcookiex12

8/19/2022

7 0

2 years ago

Amanda and her four sisters divided 1,021 stickers equally how many did each girl receive

Gemiola [76]

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 + 2

step 1

exchange the value of x for y and the value of y for x
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

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