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Anuta_ua [19.1K]

3 years ago

9

Out of every 194 containers of juice bought in a grocery store, 90 are orange juice. What fraction of juice purchased is orange

juice?

1 answer:

ipn [44]3 years ago

3 0

45/97 would be the answer to your question.

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What is the area of the figure on the coordinate plane below?

Alex787 [66]

Answer:

A 144units ^ 2

Step-by-step explanation:

5 0

3 years ago

Read 2 more answers

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 <u>c</u>ap $z=\sqrt{3-x^2-y^2}$ , 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

Costco is offering to perform the floor

const2013 [10]

Answer:

3.50x= t

Step-by-step explanation:

x square feet of teh floor

t= total cost

3 0

2 years ago

Two classes have a total of 60 students. The students need to make teams of 8t

Step2247 [10]

Answer:

We have 7 complete teams

Step-by-step explanation:

Here, we have students preparing to make a team of 8 students per team. We now want to know how many complete teams they can make if they are a total of 60;

To get this, we need the multiples of 8;

we have;

8, 16 , 24 , 32, 40 , 48 , 56

So breaking it in 8s, we have;

8 8 8 8 8 8 8

We have 7 8s;

So there would be four left overs

3 0

3 years ago

You buy milk in 1 gallon containers. One portion of waffles requires 0.4 ounces of milk. How many portions can be made with one

xenn [34]

320 portions of waffles can be made with one container.

Step-by-step explanation:

Given,

Milk is bought in 1 gallon container.

One portion of waffles require = 0.4 ounces of milk

We know that;

1 gallon = 128 ounces

Now;

No. of portions of waffles = $\frac{Total\ ounces\ of\ milk}{Milk\ used\ per\ waffle}$

No. of portions of waffles = $\frac{128}{0.4}$

No. of portions of waffles = 320

320 portions of waffles can be made with one container.

5 0

3 years ago

Read 2 more answers