Y^3 * y^5 = y^8. you add the exponents
Answer:
78 mg
Step-by-step explanation:
370 mg - 110 mg = 260 mg (amount decayed in the 8 total years)
260 mg ÷ 8 years = 32.5 mg decayed per year
32.5 × 9 years (years between 2008 and 2017) = 292.5 mg
370 mg - 292.5 mg = 77.5 mg, rounded to 78 mg
The surface (call it
) is a triangle with vertices at the points
![x=0,y=0\implies z=2\implies(0,0,2)](https://tex.z-dn.net/?f=x%3D0%2Cy%3D0%5Cimplies%20z%3D2%5Cimplies%280%2C0%2C2%29)
![x=0,y=2\implies z=-2\implies(0,2,-2)](https://tex.z-dn.net/?f=x%3D0%2Cy%3D2%5Cimplies%20z%3D-2%5Cimplies%280%2C2%2C-2%29)
![x=2,y=0\implies z=-8\implies(2,0,-8)](https://tex.z-dn.net/?f=x%3D2%2Cy%3D0%5Cimplies%20z%3D-8%5Cimplies%282%2C0%2C-8%29)
Parameterize
by
![\vec s(u,v)=(1-v)(2,0,-8)+v\bigg((1-u)(0,2,-2)+u(0,0,2)\bigg)=(2-2v,2v-2uv,-8+6v+4uv)](https://tex.z-dn.net/?f=%5Cvec%20s%28u%2Cv%29%3D%281-v%29%282%2C0%2C-8%29%2Bv%5Cbigg%28%281-u%29%280%2C2%2C-2%29%2Bu%280%2C0%2C2%29%5Cbigg%29%3D%282-2v%2C2v-2uv%2C-8%2B6v%2B4uv%29)
with
and
. Take the normal vector to
to be
![\vec s_v\times\vec s_u=(20v,8v,4v)](https://tex.z-dn.net/?f=%5Cvec%20s_v%5Ctimes%5Cvec%20s_u%3D%2820v%2C8v%2C4v%29)
Then the flux of
across
is
![\displaystyle\iint_S\vec F(x,y,z)\cdot\mathrm d\vec S=\int_0^1\int_0^1\vec F(x(u,v),y(u,v),z(u,v))\cdot(\vec s_v\times\vec s_u)\,\mathrm du\,\mathrm dv](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Ciint_S%5Cvec%20F%28x%2Cy%2Cz%29%5Ccdot%5Cmathrm%20d%5Cvec%20S%3D%5Cint_0%5E1%5Cint_0%5E1%5Cvec%20F%28x%28u%2Cv%29%2Cy%28u%2Cv%29%2Cz%28u%2Cv%29%29%5Ccdot%28%5Cvec%20s_v%5Ctimes%5Cvec%20s_u%29%5C%2C%5Cmathrm%20du%5C%2C%5Cmathrm%20dv)
![=\displaystyle\int_0^1\int_0^1(6-6v,2v-2uv,-8+6v+4uv)\cdot(20v,8v,4v)\,\mathrm du\,\mathrm dv](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle%5Cint_0%5E1%5Cint_0%5E1%286-6v%2C2v-2uv%2C-8%2B6v%2B4uv%29%5Ccdot%2820v%2C8v%2C4v%29%5C%2C%5Cmathrm%20du%5C%2C%5Cmathrm%20dv)
![=\displaystyle8\int_0^1\int_0^1(11v-10v^2)\,\mathrm du\,\mathrm dv=\boxed{\frac{52}3}](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle8%5Cint_0%5E1%5Cint_0%5E1%2811v-10v%5E2%29%5C%2C%5Cmathrm%20du%5C%2C%5Cmathrm%20dv%3D%5Cboxed%7B%5Cfrac%7B52%7D3%7D)