This one is mildish please hepl

If the company is planning to produce 90,000 containers of crunchy peanut butter

Sergeu [11.5K]

Is there more to the problem?

3 0

4 years ago

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 histogram represents the data set with the smallest standard deviation? A histogram titled Squad 1 has x-ring hits on the

Delicious77 [7]

The histogram which represents the data set with the smallest standard deviation is: C. Squad 3.

<h3>What is a histogram?</h3>

A histogram can be defined as a type of chart that's used to graphically represent a set of data points into user-specified ranges, especially through the use of rectangular bars.

Basically, standard deviation is a statistical tool which can be used to determine the measure of spread for the data represented in a histogram, especially based on the frequency of the specified ranges.

In this scenario, the histogram which represents the data set with the smallest standard deviation is Squad 3 because it has the least frequency on the y-axis.

Read more on histogram here: brainly.com/question/21304143

#SPJ1

7 0

2 years ago

Read 2 more answers

What is the slope of a line that is perpendicular to a line whose slope is -1/3.

solniwko [45]

Answer:

3

Step-by-step explanation:

the perpendicular slope is the negative reciprocal.

-1 divided by -1/3 = 3

6 0

3 years ago

Identify the angle as adjacent, vertical, complimentary, supplementary, congruent, or linear pair

tatuchka [14]

Answer:

complimemtary

Step-by-step explanation:

3 0

3 years ago