The compatible number is 7 (seven)
The 8 in 683 is in the tens place value, making it worth 80units while the 8 in 48 is in the ones place value making it worth 8units.
The new area would be 602,000 yd².
When you enlarge a shape (or a park) by a factor, you are multiplying the length of its sides by that factor. In this case, we are multiplying its length and width by 4.
Let l be the old length and w be the new length. The old area would be given by A=lw.
Enlarging this by a factor of 4 would make the new length 4l and the new width 4w. The new area would be 4l(4w)=16lw. The new area is 16 times larger than the old area, so:
37,625(16)=602,000.
Each box = $6.
The letter b represents one box.
The letter d represents the number of dollars spent on boxes.
The equation that best describes the situation at hand is
d = 6b
1. By the chain rule,
![\dfrac{\mathrm dz}{\mathrm dt}=\dfrac{\partial z}{\partial x}\dfrac{\mathrm dx}{\mathrm dt}+\dfrac{\partial z}{\partial y}\dfrac{\mathrm dy}{\mathrm dt}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dz%7D%7B%5Cmathrm%20dt%7D%3D%5Cdfrac%7B%5Cpartial%20z%7D%7B%5Cpartial%20x%7D%5Cdfrac%7B%5Cmathrm%20dx%7D%7B%5Cmathrm%20dt%7D%2B%5Cdfrac%7B%5Cpartial%20z%7D%7B%5Cpartial%20y%7D%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dt%7D)
I'm going to switch up the notation to save space, so for example,
is shorthand for
.
![z_t=z_xx_t+z_yy_t](https://tex.z-dn.net/?f=z_t%3Dz_xx_t%2Bz_yy_t)
We have
![x=e^{-t}\implies x_t=-e^{-t}](https://tex.z-dn.net/?f=x%3De%5E%7B-t%7D%5Cimplies%20x_t%3D-e%5E%7B-t%7D)
![y=e^t\implies y_t=e^t](https://tex.z-dn.net/?f=y%3De%5Et%5Cimplies%20y_t%3De%5Et)
![z=\tan(xy)\implies\begin{cases}z_x=y\sec^2(xy)=e^t\sec^2(1)\\z_y=x\sec^2(xy)=e^{-t}\sec^2(1)\end{cases}](https://tex.z-dn.net/?f=z%3D%5Ctan%28xy%29%5Cimplies%5Cbegin%7Bcases%7Dz_x%3Dy%5Csec%5E2%28xy%29%3De%5Et%5Csec%5E2%281%29%5C%5Cz_y%3Dx%5Csec%5E2%28xy%29%3De%5E%7B-t%7D%5Csec%5E2%281%29%5Cend%7Bcases%7D)
![\implies z_t=e^t\sec^2(1)(-e^{-t})+e^{-t}\sec^2(1)e^t=0](https://tex.z-dn.net/?f=%5Cimplies%20z_t%3De%5Et%5Csec%5E2%281%29%28-e%5E%7B-t%7D%29%2Be%5E%7B-t%7D%5Csec%5E2%281%29e%5Et%3D0)
Similarly,
![w_t=w_xx_t+w_yy_t+w_zz_t](https://tex.z-dn.net/?f=w_t%3Dw_xx_t%2Bw_yy_t%2Bw_zz_t)
where
![x=\cosh^2t\implies x_t=2\cosh t\sinh t](https://tex.z-dn.net/?f=x%3D%5Ccosh%5E2t%5Cimplies%20x_t%3D2%5Ccosh%20t%5Csinh%20t)
![y=\sinh^2t\implies y_t=2\cosh t\sinh t](https://tex.z-dn.net/?f=y%3D%5Csinh%5E2t%5Cimplies%20y_t%3D2%5Ccosh%20t%5Csinh%20t)
![z=t\implies z_t=1](https://tex.z-dn.net/?f=z%3Dt%5Cimplies%20z_t%3D1)
To capture all the partial derivatives of
, compute its gradient:
![\nabla w=\langle w_x,w_y,w_z\rangle=\dfrac{\langle1,-1,1\rangle}{\sqrt{1-(x-y+z)^2}}}=\dfrac{\langle1,-1,1\rangle}{\sqrt{-2t-t^2}}](https://tex.z-dn.net/?f=%5Cnabla%20w%3D%5Clangle%20w_x%2Cw_y%2Cw_z%5Crangle%3D%5Cdfrac%7B%5Clangle1%2C-1%2C1%5Crangle%7D%7B%5Csqrt%7B1-%28x-y%2Bz%29%5E2%7D%7D%7D%3D%5Cdfrac%7B%5Clangle1%2C-1%2C1%5Crangle%7D%7B%5Csqrt%7B-2t-t%5E2%7D%7D)
![\implies w_t=\dfrac1{\sqrt{-2t-t^2}}](https://tex.z-dn.net/?f=%5Cimplies%20w_t%3D%5Cdfrac1%7B%5Csqrt%7B-2t-t%5E2%7D%7D)
2. The problem is asking for
and
. But
is already a function of
, so the chain rule isn't needed here. I suspect it's supposed to say "find
and
" instead.
If that's the case, then
![z_s=z_xx_s+z_yy_s](https://tex.z-dn.net/?f=z_s%3Dz_xx_s%2Bz_yy_s)
![z_t=z_xx_t+z_yy_t](https://tex.z-dn.net/?f=z_t%3Dz_xx_t%2Bz_yy_t)
as the hint suggests. We have
![z=\sin x\cos y\implies\begin{cases}z_x=\cos x\cos y=\cos(s+t)\cos(s^2t)\\z_y=-\sin x\sin y=-\sin(s+t)\sin(s^2t)\end{cases}](https://tex.z-dn.net/?f=z%3D%5Csin%20x%5Ccos%20y%5Cimplies%5Cbegin%7Bcases%7Dz_x%3D%5Ccos%20x%5Ccos%20y%3D%5Ccos%28s%2Bt%29%5Ccos%28s%5E2t%29%5C%5Cz_y%3D-%5Csin%20x%5Csin%20y%3D-%5Csin%28s%2Bt%29%5Csin%28s%5E2t%29%5Cend%7Bcases%7D)
![x=s+t\implies x_s=x_t=1](https://tex.z-dn.net/?f=x%3Ds%2Bt%5Cimplies%20x_s%3Dx_t%3D1)
![y=s^2t\implies\begin{cases}y_s=2st\\y_t=s^2\end{cases}](https://tex.z-dn.net/?f=y%3Ds%5E2t%5Cimplies%5Cbegin%7Bcases%7Dy_s%3D2st%5C%5Cy_t%3Ds%5E2%5Cend%7Bcases%7D)
Putting everything together, we get
![z_s=\cos(s+t)\cos(s^2t)-2st\sin(s+t)\sin(s^2t)](https://tex.z-dn.net/?f=z_s%3D%5Ccos%28s%2Bt%29%5Ccos%28s%5E2t%29-2st%5Csin%28s%2Bt%29%5Csin%28s%5E2t%29)
![z_t=\cos(s+t)\cos(s^2t)-s^2\sin(s+t)\sin(s^2t)](https://tex.z-dn.net/?f=z_t%3D%5Ccos%28s%2Bt%29%5Ccos%28s%5E2t%29-s%5E2%5Csin%28s%2Bt%29%5Csin%28s%5E2t%29)