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kvv77 [185]
3 years ago
9

How would I solve for this?

Mathematics
1 answer:
Mariulka [41]3 years ago
7 0
I couldn’t read your hand writing I hope that was a z...not a 2...anyway the explanation is that you use a calculator

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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
Which of the following is equivalent to 75/4 <br> A- 18 3/18<br> B- 18 3/4<br> C- 71<br> D- 79
sattari [20]
B.
4 goes into 75 18 times and has a remainder of 3, so it’s 18 3/4
5 0
3 years ago
Read 2 more answers
What is the correlation coefficient for the data shown in the table?
xxTIMURxx [149]

Answer:

The correlation coefficient is -1

Step-by-step explanation:

Here, we want to get the correlation coefficient for the given dataset

from the table, what do we observe?

As x is increasing by 5 units;

y is decreasing by 5 units

What this mean is that there is a negative correlation between both

Thus, what we have here is that, since the rate of increase/decrease on both ends is of equal magnitude but opposite signs, we can conclude that the correlation coefficient is -1

8 0
2 years ago
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Almost Pinejuice (y) costs one quarter as much as pineapple juice (p) does. What equation represents this statement?
galben [10]
The answer is D since 1/4 of the pineapple juice will give you the pinejuice
6 0
3 years ago
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A person invests $1,150 in an account that earns 5% annual interest compounded continuously. Find when the value of the investme
AfilCa [17]

Answer:

11.1 years

Step-by-step explanation:

The formula for interest compounding continuously is:

A(t)=Pe^{rt}

Where A(t) is the amount after the compounding, P is the initial deposit, r is the interest rate in decimal form, and t is the time in years.  Filling in what we have looks like this:

2000=1150e^{{.05t}

We will simplify this first a bit by dividing 2000 by 1150 to get

1.739130435=e^{.05t}

To get that t out the exponential position it is currently in we have to take the natural log of both sides.  Since a natural log has a base of e, taking the natual log of e cancels both of them out.  They "undo" each other, for lack of a better way to explain it.  That leaves us with

ln(1.739130435)=.05t

Taking the natural log of that decimal on our calculator gives us

.5533852383=.05t

Now divide both sides by .05 to get t = 11.06770477 which rounds to 11.1 years.

7 0
2 years ago
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