Answer:
It would be ASA
Step-by-step explanation:
Those two triangles are congregant so it would be A. ASA
The ODE
![M(x,y)\,\mathrm dx+N(x,y)\,\mathrm dy=0](https://tex.z-dn.net/?f=M%28x%2Cy%29%5C%2C%5Cmathrm%20dx%2BN%28x%2Cy%29%5C%2C%5Cmathrm%20dy%3D0)
is exact if
![\dfrac{\partial M}{\partial y}=\dfrac{\partial N}{\partial x}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20M%7D%7B%5Cpartial%20y%7D%3D%5Cdfrac%7B%5Cpartial%20N%7D%7B%5Cpartial%20x%7D)
We have
![M=-xy\sin x+2y\cos x\implies M_y=-x\sin x+2\cos x](https://tex.z-dn.net/?f=M%3D-xy%5Csin%20x%2B2y%5Ccos%20x%5Cimplies%20M_y%3D-x%5Csin%20x%2B2%5Ccos%20x)
![N=2x\cos x\implies N_x=2\cos x-2x\sin x](https://tex.z-dn.net/?f=N%3D2x%5Ccos%20x%5Cimplies%20N_x%3D2%5Ccos%20x-2x%5Csin%20x)
so the ODE is indeed not exact.
Multiplying both sides of the ODE by
gives
![\mu M=-x^2y^2\sin x+2xy^2\cos x\implies(\mu M)_y=-2x^2y\sin x+4xy\cos x](https://tex.z-dn.net/?f=%5Cmu%20M%3D-x%5E2y%5E2%5Csin%20x%2B2xy%5E2%5Ccos%20x%5Cimplies%28%5Cmu%20M%29_y%3D-2x%5E2y%5Csin%20x%2B4xy%5Ccos%20x)
![\mu N=2x^2y\cos x\implies(\mu N)_x=4xy\cos x-2x^2y\sin x](https://tex.z-dn.net/?f=%5Cmu%20N%3D2x%5E2y%5Ccos%20x%5Cimplies%28%5Cmu%20N%29_x%3D4xy%5Ccos%20x-2x%5E2y%5Csin%20x)
so that
, and the modified ODE is exact.
We're looking for a solution of the form
![\Psi(x,y)=C](https://tex.z-dn.net/?f=%5CPsi%28x%2Cy%29%3DC)
so that by differentiation, we should have
![\Psi_x\,\mathrm dx+\Psi_y\,\mathrm dy=0](https://tex.z-dn.net/?f=%5CPsi_x%5C%2C%5Cmathrm%20dx%2B%5CPsi_y%5C%2C%5Cmathrm%20dy%3D0)
![\implies\begin{cases}\Psi_x=\mu M\\\Psi_y=\mu N\end{cases}](https://tex.z-dn.net/?f=%5Cimplies%5Cbegin%7Bcases%7D%5CPsi_x%3D%5Cmu%20M%5C%5C%5CPsi_y%3D%5Cmu%20N%5Cend%7Bcases%7D)
Integrating both sides of the second equation with respect to
gives
![\Psi_y=2x^2y\cos x\implies\Psi=x^2y^2\cos x+f(x)](https://tex.z-dn.net/?f=%5CPsi_y%3D2x%5E2y%5Ccos%20x%5Cimplies%5CPsi%3Dx%5E2y%5E2%5Ccos%20x%2Bf%28x%29)
Differentiating both sides with respect to
gives
![\Psi_x=-x^2y^2\sin x+2xy^2\cos x=2xy^2\cos x-x^2y^2\sin x+\dfrac{\mathrm df}{\mathrm dx}](https://tex.z-dn.net/?f=%5CPsi_x%3D-x%5E2y%5E2%5Csin%20x%2B2xy%5E2%5Ccos%20x%3D2xy%5E2%5Ccos%20x-x%5E2y%5E2%5Csin%20x%2B%5Cdfrac%7B%5Cmathrm%20df%7D%7B%5Cmathrm%20dx%7D)
![\implies\dfrac{\mathrm df}{\mathrm dx}=0\implies f(x)=c](https://tex.z-dn.net/?f=%5Cimplies%5Cdfrac%7B%5Cmathrm%20df%7D%7B%5Cmathrm%20dx%7D%3D0%5Cimplies%20f%28x%29%3Dc)
for some constant
.
So the general solution to this ODE is
![x^2y^2\cos x+c=C](https://tex.z-dn.net/?f=x%5E2y%5E2%5Ccos%20x%2Bc%3DC)
or simply
![x^2y^2\cos x=C](https://tex.z-dn.net/?f=x%5E2y%5E2%5Ccos%20x%3DC)
4/5=.80
4/.80= 5
5*7= 35
You can make 35 granola bars with 4 cups of chocolate chips.
I hope this helps!
~kaikers
$75 if you do guess and check with random numbers multiplying them by .6 (60% remaining) you will eventually find the answer