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2 years ago

Please answer

Mathematics

2 answers:

murzikaleks [220]2 years ago

7 0

The correct answer is A. 2,432

adell [148]2 years ago

4 0

Answer:

the correct answer is 2,432

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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 cap $z=\sqrt{3-x^2-y^2}$ , 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

X + y = 152, 8.5x + 12y = 1,656 How many hats were sold?

Elodia [21]

Answer:

x = 48 and y = 104

Step-by-step explanation:

Given equations are:

$x+y = 152\\8.5x+12y=1656$

From equation 1:

$x = 152-y$

Putting the value of y in equation 2

$8.5(152-y)+12y = 1656\\1292-8.5y+12y = 1656\\3.5y+1292 = 1656\\3.5y = 1656-1292\\3.5y = 364\\\frac{3.5y}{3.5} = \frac{364}{3.5}\\y = 104$

Now we have to put the value of y in one of the equation to find the value of x

Putting y = 104 in the first equation

$x+y = 152\\x+ 104 = 152\\x = 152-104\\x = 48$

Hence,

The solution of the system of equations is x = 48 and y = 104

The value of variable which was assumed for number of hats, is the total number of hats.

6 0

2 years ago

Plz help me. I forgot how to do this XD

nirvana33 [79]

This question has to do with making an approximation based on the diagram of the angle. Angle KLM is an obtuse angle meaning it is larger then 90° but is also less than 180°. The only choices that lie within this range are choices B and D. However, it is safe to assume that angle KLM is closer to 180° than to 90° based on the diagram, therefore the answer must be choice B.

I hope this helps.

5 0

3 years ago

What’s the equation ? (2,-2) ( 4,1)

oksano4ka [1.4K]

Answer:

y = 3 2x − 5

Step-by-step explanation:

5 0

3 years ago

A pizza parlor sold 38 pizzas during a dinner hour. If each pizza contained 8 slices, how many slices of pizza were sold?

AlladinOne [14]

38 x 8 = 304
So the answer is b