Answer:
Least: -s
Greatest: s
closest to 0: s+r
Step-by-step explanatio :
It's kind of obvious that the smallest value is going to equal -s which means that the largest is s which leaves s+r to be the value that's cloest to 0
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)
15)
∠EAD ≅ ∠CAB . . . . . . reflexive property. An angle is congruent with itself.
∠AED ≅ ∠ACB . . . . . . given
ΔABC ≅ ΔADE . . . . . . angle-angle similarity: If two angles are congruent, all three angles are congruent and the triangles are similar.
Answer:
£112
Step-by-step explanation:
<u>Sushi Fiona</u>
3 : 5
Sushi gets £<u>42</u> ---- > 42/3 = 14 -----> 14x5 = £<u>70 (Fiona)</u>
42 + 70 = £112
15/72=5/24
Divide both by three.
:)