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1 year ago

Solve 5x+4y= 9 for y.

Mathematics

1 answer:

natima [27]1 year ago

8 0

Given:

$5x+4y=9$

To solve for y:

Explanation:

Isolating 4y, we get,

$4y=9-5x$

Dividing by 4 into both sides, we get

$y=\frac{9-5x}{4}$

Final answer:

$y=\frac{9-5x}{4}$

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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

kifflom [539]

Looks like we have

$\vec F(x,y,z)=z^2x\,\vec\imath+\left(\dfrac{y^3}3+\sin z\right)\,\vec\jmath+(x^2z+y^2)\,\vec k$

which has divergence

$\nabla\cdot\vec F(x,y,z)=\dfrac{\partial(z^2x)}{\partial x}+\dfrac{\partial\left(\frac{y^3}3+\sin z\right)}{\partial y}+\dfrac{\partial(x^2z+y^2)}{\partial z}=z^2+y^2+x^2$

By the divergence theorem, the integral of $\vec F$ across $S$ is equal to the integral of $\nabla\cdot\vec F$ over $R$ , where $R$ is the region enclosed by $S$ . Of course, $S$ is not a closed surface, but we can make it so by closing off the hemisphere $S$ by attaching it to the disk $x^2+y^2\le1$ (call it $D$ ) so that $R$ has boundary $S\cup D$ .

Then by the divergence theorem,

$\displaystyle\iint_{S\cup D}\vec F\cdot\mathrm d\vec S=\iiint_R(x^2+y^2+z^2)\,\mathrm dV$

Compute the integral in spherical coordinates, setting

$\begin{cases}x=\rho\cos\theta\sin\varphi\\y=\rho\sin\theta\sin\varphi\\z=\rho\cos\varphi\end{cases}\implies\mathrm dV=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi$

so that the integral is

$\displaystyle\iiint_R(x^2+y^2+z^2)\,\mathrm dV=\int_0^{\pi/2}\int_0^{2\pi}\int_0^1\rho^4\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi=\frac{2\pi}5$

The integral of $\vec F$ across $S\cup D$ is equal to the integral of $\vec F$ across $S$ plus the integral across $D$ (without outward orientation, so that

$\displaystyle\iint_S\vec F\cdot\mathrm d\vec S=\frac{2\pi}5-\iint_D\vec F\cdot\mathrm d\vec S$

Parameterize $D$ by

$\vec s(u,v)=u\cos v\,\vec\imath+u\sin v\,\vec\jmath$

with $0\le u\le1$ and $0\le v\le2\pi$ . Take the normal vector to $D$ to be

$\dfrac{\partial\vec s}{\partial v}\times\dfrac{\partial\vec s}{\partial u}=-u\,\vec k$

Then we have

$\displaystyle\iint_D\vec F\cdot\mathrm d\vec S=\int_0^{2\pi}\int_0^1\left(\frac{u^3}3\sin^3v\,\vec\jmath+u^2\sin^2v\,\vec k\right)\times(-u\,\vec k)\,\mathrm du\,\mathrm dv$

$=\displaystyle-\int_0^{2\pi}\int_0^1u^3\sin^2v\,\mathrm du\,\mathrm dv=-\frac\pi4$

Finally,

$\displaystyle\iint_S\vec F\cdot\mathrm d\vec S=\frac{2\pi}5-\left(-\frac\pi4\right)=\boxed{\frac{13\pi}{20}}$

6 0

4 years ago

Find the sum or difference.write your answer in simplest form 2/9+1/3=

lukranit [14]

Answer:

7/9

Step-by-step explanation:

7 0

3 years ago

Read 2 more answers

The plane is vertical. What best describes the shape of the cross section

frutty [35]

Answer:

triangle

Step-by-step explanation:

Any cut plane parallel to the parallel faces will create a cross section congruent with the parallel faces. The faces are triangles, so the cross section is a triangle.

3 0

3 years ago

I don’t know how to solve things like this

EleoNora [17]

To do this problem you need to find angle adc which is a part of angle BDA. angle BDA is 90 degrees so the 2 parts must add up to 90 degrees so you set up the problem like this
-2x + 144 - 3x +51 =90
then you combine like terms
-5x +195 = 90
then you subtract 195 from 90
-5x = -105
then you divide -5 by -5 and -105 by -5 and get
x=21
then you plug x in and get your angle measure
-3(21) +51
then you get the angle measure as 114 degrees.

8 0

3 years ago

Some trapezoid has an area of 213.86 yd2 and a height of 18 yd. If one of the bases measures 5 yd, then the other base must meas

Dafna1 [17]

Answer:

The other base measure <u>18.76</u> yd.

Step-by-step explanation:

A = h ((a+b) / 2)

213.86 = 18 ((5 + b) /2)

(5 + b) /2 = 11.88

5 + b = 23.76

7 0

3 years ago