25 hundredths in simplest form

Mathematics

2 answers:

sergiy2304 [10]3 years ago

6 0

The answer is 1/4 or .25

IRISSAK [1]3 years ago

5 0

.25 or 1/4 Is the simplest form

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Let f(x,y,z) = ztan-1(y2) i + z3ln(x2 + 1) j + z k. find the flux of f across the part of the paraboloid x2 + y2 + z = 3 that li

Sophie [7]

Consider the closed region $V$ bounded simultaneously by the paraboloid and plane, jointly denoted $S$ . By the divergence theorem,

$\displaystyle\iint_S\mathbf f(x,y,z)\cdot\mathrm dS=\iiint_V\nabla\cdot\mathbf f(x,y,z)\,\mathrm dV$

And since we have

$\nabla\cdot\mathbf f(x,y,z)=1$

the volume integral will be much easier to compute. Converting to cylindrical coordinates, we have

$\displaystyle\iiint_V\nabla\cdot\mathbf f(x,y,z)\,\mathrm dV=\iiint_V\mathrm dV$
$=\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=1}\int_{z=2}^{z=3-r^2}r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta$
$=\displaystyle2\pi\int_{r=0}^{r=1}r(3-r^2-2)\,\mathrm dr$
$=\dfrac\pi2$

Then the integral over the paraboloid would be the difference of the integral over the total surface and the integral over the disk. Denoting the disk by $D$ , we have

$\displaystyle\iint_{S-D}\mathbf f\cdot\mathrm dS=\frac\pi2-\iint_D\mathbf f\cdot\mathrm dS$

Parameterize $D$ by

$\mathbf s(u,v)=u\cos v\,\mathbf i+u\sin v\,\mathbf j+2\,\mathbf k$
$\implies\mathbf s_u\times\mathbf s_v=u\,\mathbf k$

which would give a unit normal vector of $\mathbf k$ . However, the divergence theorem requires that the closed surface $S$ be oriented with outward-pointing normal vectors, which means we should instead use $\mathbf s_v\times\mathbf s_u=-u\,\mathbf k$ .

Now,

$\displaystyle\iint_D\mathbf f\cdot\mathrm dS=\int_{u=0}^{u=1}\int_{v=0}^{v=2\pi}\mathbf f(x(u,v),y(u,v),z(u,v))\cdot(-u\,\mathbf k)\,\mathrm dv\,\mathrm du$
$=\displaystyle-4\pi\int_{u=0}^{u=1}u\,\mathrm du$
$=-2\pi$

So, the flux over the paraboloid alone is

$\displaystyle\iint_{S-D}\mathbf f\cdot\mathrm dS=\frac\pi2-(-2\pi)=\dfrac{5\pi}2$

6 0

3 years ago

What equals 1050 in multiplication?

kakasveta [241]

There is multiple possibilities:

1050 equals 1 times 1050

1050 equals 2 times 525

1050 equals 3 times 350

1050 equals 5 times 210

1050 equals 6 times 175

1050 equals 7 times 150

1050 equals 10 times 105

1050 equals 14 times 75

1050 equals 15 times 70

1050 equals 21 times 50

1050 equals 25 times 42

1050 equals 30 times 35

1050 equals 35 times 30

1050 equals 42 times 25

1050 equals 50 times 21

1050 equals 70 times 15

1050 equals 75 times 14

1050 equals 105 times 10

1050 equals 150 times 7

1050 equals 175 times 6

1050 equals 210 times 5

1050 equals 350 times 3

1050 equals 525 times 2

1050 equals 1050 times 1

8 0

3 years ago

Read 2 more answers

Solve the equation Square root of x minus 5+ 7 = 11 for the variable. Show each step of your solution process. (10 points)

Murrr4er [49]

$\sqrt{x - 5} + 7 = 11$