Given:
There is a ratio given as 16:9 of width to height and diagonal is 27 iniches
Required:
We need to find the value of height
Explanation:
By ratio
![\begin{gathered} \frac{w}{h}=\frac{16}{9} \\ \\ h=\frac{9}{16}w \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%5Cfrac%7Bw%7D%7Bh%7D%3D%5Cfrac%7B16%7D%7B9%7D%20%5C%5C%20%20%5C%5C%20h%3D%5Cfrac%7B9%7D%7B16%7Dw%20%5Cend%7Bgathered%7D)
where w is width and h is height
by using pythagorean theorem
![\begin{gathered} 27^2=h^2+w^2 \\ 729=\frac{81}{256}w^2+w^2 \\ \\ 186624=81w^2+256w^2 \\ w^2=553.78 \\ w=23.5\text{ in} \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%2027%5E2%3Dh%5E2%2Bw%5E2%20%5C%5C%20729%3D%5Cfrac%7B81%7D%7B256%7Dw%5E2%2Bw%5E2%20%5C%5C%20%20%5C%5C%20186624%3D81w%5E2%2B256w%5E2%20%5C%5C%20w%5E2%3D553.78%20%5C%5C%20w%3D23.5%5Ctext%7B%20in%7D%20%5Cend%7Bgathered%7D)
to find h
![h=\frac{9}{16}*23.5=13.24\text{ in}](https://tex.z-dn.net/?f=h%3D%5Cfrac%7B9%7D%7B16%7D%2A23.5%3D13.24%5Ctext%7B%20in%7D)
Final answer:
height is 13.24 inches
![\vec f(x,y,z)=y\,\vec\imath-4yz\,\vec\jmath+3z^2\,\vec k](https://tex.z-dn.net/?f=%5Cvec%20f%28x%2Cy%2Cz%29%3Dy%5C%2C%5Cvec%5Cimath-4yz%5C%2C%5Cvec%5Cjmath%2B3z%5E2%5C%2C%5Cvec%20k)
![\implies\nabla\cdot\vec f(x,y,z)=0-4z+6z=2z](https://tex.z-dn.net/?f=%5Cimplies%5Cnabla%5Ccdot%5Cvec%20f%28x%2Cy%2Cz%29%3D0-4z%2B6z%3D2z)
By the divergence theorem,
![\displaystyle\iint_{\partial W}\vec f\cdot\mathrm d\vec S=\iiint_W2z\,\mathrm dV](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Ciint_%7B%5Cpartial%20W%7D%5Cvec%20f%5Ccdot%5Cmathrm%20d%5Cvec%20S%3D%5Ciiint_W2z%5C%2C%5Cmathrm%20dV)
I'll assume a sphere of radius
centered at the origin, and that
is bounded below by the plane
. Convert to spherical coordinates, taking
![x=\rho\cos\theta\sin\varphi](https://tex.z-dn.net/?f=x%3D%5Crho%5Ccos%5Ctheta%5Csin%5Cvarphi)
![y=\rho\sin\theta\sin\varphi](https://tex.z-dn.net/?f=y%3D%5Crho%5Csin%5Ctheta%5Csin%5Cvarphi)
![z=\rho\cos\varphi](https://tex.z-dn.net/?f=z%3D%5Crho%5Ccos%5Cvarphi)
Then
![\displaystyle\iiint_W2z\,\mathrm dV=\int_0^{\pi/2}\int_0^{2\pi}\int_0^r2\rho^3\cos\varphi\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi=\pi r^4](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Ciiint_W2z%5C%2C%5Cmathrm%20dV%3D%5Cint_0%5E%7B%5Cpi%2F2%7D%5Cint_0%5E%7B2%5Cpi%7D%5Cint_0%5Er2%5Crho%5E3%5Ccos%5Cvarphi%5Csin%5Cvarphi%5C%2C%5Cmathrm%20d%5Crho%5C%2C%5Cmathrm%20d%5Ctheta%5C%2C%5Cmathrm%20d%5Cvarphi%3D%5Cpi%20r%5E4)