The square (call it
) has one vertex at the origin (0, 0, 0) and one edge on the y-axis, which tells us another vertex is (0, 3, 0). The normal vector to the plane is
, which is enough information to figure out the equation of the plane containing
:
![(x\,\vec\imath+y\,\vec\jmath+z\,\vec k)\cdot(\vec\imath-\vec k)=0\implies x-z=0\implies z=x](https://tex.z-dn.net/?f=%28x%5C%2C%5Cvec%5Cimath%2By%5C%2C%5Cvec%5Cjmath%2Bz%5C%2C%5Cvec%20k%29%5Ccdot%28%5Cvec%5Cimath-%5Cvec%20k%29%3D0%5Cimplies%20x-z%3D0%5Cimplies%20z%3Dx)
We can parameterize this surface by
![\vec s(x,y)=x\,\vec\imath+y\,\vec\jmath+x\,\vec k](https://tex.z-dn.net/?f=%5Cvec%20s%28x%2Cy%29%3Dx%5C%2C%5Cvec%5Cimath%2By%5C%2C%5Cvec%5Cjmath%2Bx%5C%2C%5Cvec%20k)
for
and
. Then the flux of
, assumed to be
,
is
![\displaystyle\iint_S\vec F(x,y,z)\cdot\mathrm d\vec S=\iint_S\vec F(\vec s(x,y))\cdot\vec n\,\mathrm dx\,\mathrm dy](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Ciint_S%5Cvec%20F%28x%2Cy%2Cz%29%5Ccdot%5Cmathrm%20d%5Cvec%20S%3D%5Ciint_S%5Cvec%20F%28%5Cvec%20s%28x%2Cy%29%29%5Ccdot%5Cvec%20n%5C%2C%5Cmathrm%20dx%5C%2C%5Cmathrm%20dy)
![=\displaystyle\int_0^3\int_0^{3/\sqrt2}\left((4+e^{xy}+9x)\,\vec\imath+(9+e^{xy}+4x)\,\vec\jmath+(e^{xy}+9x)\,\vec k\right)\cdot(\vec\imath-\vec k)\,\mathrm dx\,\mathrm dy](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle%5Cint_0%5E3%5Cint_0%5E%7B3%2F%5Csqrt2%7D%5Cleft%28%284%2Be%5E%7Bxy%7D%2B9x%29%5C%2C%5Cvec%5Cimath%2B%289%2Be%5E%7Bxy%7D%2B4x%29%5C%2C%5Cvec%5Cjmath%2B%28e%5E%7Bxy%7D%2B9x%29%5C%2C%5Cvec%20k%5Cright%29%5Ccdot%28%5Cvec%5Cimath-%5Cvec%20k%29%5C%2C%5Cmathrm%20dx%5C%2C%5Cmathrm%20dy)
![=\displaystyle\int_0^3\int_0^{3/\sqrt2}4\,\mathrm dx\,\mathrm dy=\boxed{18\sqrt2}](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle%5Cint_0%5E3%5Cint_0%5E%7B3%2F%5Csqrt2%7D4%5C%2C%5Cmathrm%20dx%5C%2C%5Cmathrm%20dy%3D%5Cboxed%7B18%5Csqrt2%7D)
Answer:
17/12
Step-by-step explanation:
Answer:
Unit rate is 3 ounce of water per ounce of mix
Step-by-step explanation:
you combine 12 ounces of mix with 36 ounces of water.
12 ounces of mix for 36 ounces of water
For unit we need to find the ounces of water for 1 ounce of mix
12 ounces of mix for 36 ounces of water
so 1 ounces of mix for 36 / 12 = 3 ounces of water
3 ounces of water for 1 ounce of mix
Hence, 3 ounces of water per ounce of juice
Ok, so first this is a system. You have:
8x+12y=3200
And
X+y=300.
That means y= -X + 300.
If you replace "y" in the first equation with -X+300, it would be 8x+12(-X+300)=3200.
If you simplify that, X will be equal to 100. Now go back to the second equation, which is X+y= 300.
Now that you know X is 100, you can subtract it from 300 to find y.
So,
X=100 and y=200!
Hope this helps!!!
Answer: She can make 8 bracelets that are 11.43 centimeters.
Step-by-step explanation: Since one yard is equal to 91.44 centimeters you can divide this by 8, the amount of bracelets, and you get 11.43 centimeters