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3 years ago

Y-1=2/3(x+9) in standard form

1 answer:

3 0

First, distribute 2/3 to the values in the parentheses. This gives you y-1=2/3x+6.

To get y alone, we add 1 to both sides. This gets you y=2/3x+7.

y=mx+b
y=2/3x+7

It matches the standard form, so your final answer would be y=2/3x+7.

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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 <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

There are 20 nickels in one dollar. Write an equation to find the number of nickels n in any number of dollars d

n200080 [17]

5+5+5+5+5+5+5+5+5+5+5+5+5+5+5+5+5+5+5+5

20 nickles

20•5=1.00

4 0

2 years ago

7 - 3x = 5x + 13. <br><br> What is x?

eduard

Answer:

x= -3/4 or -0.75

Step-by-step explanation:

1) subtract 7 from both sides which would make it

-3x= 5x+6

2) Subtract 5x from both sides which would make it

-8x-6

3) Divide both sides by -8

which gives you -3/4

7 0

2 years ago

Read 2 more answers

What is the slope of the line represented by the equation y = x – 3?

leonid [27]

I THINK ITS C GOODLUCK

7 0

2 years ago

A sheet of standard-size copy paper measures 8.5 in. by 11 in. If a ream (500 sheets) of this paper has a volume of 187(to the t

Alexxandr [17]

<h3>Answer: 2 inches</h3>

============================================================

Explanation:

The area of any given sheet of paper is 8.5*11 = 93.5 square inches.

Divide the volume over this area to get the height of the stack

187/(93.5) = 2

The stack is 2 inches tall.

Extra info: The thickness of each piece of paper is 2/500 = 0.004 of an inch which approximates to about 0.1016 mm.

3 0

2 years ago