Let Q = quarters and D = dimes
There are a total of 1202 coins, so we have to add the quarters and dimes together to get this.
Q + D = 1202
Quarters have a value of 25 cents or 0.25, dimes have a value of 10 cents or 0.10. The combination of the amount of quarters and their respective value and the amount of dimes and their respective value equate to $123.50.
0.25Q + 0.10D = 123.50
Put the two equations together. Your system of equations is:
Q + D = 1202
0.25Q + 0.10D = 123.50
What are
![p](https://tex.z-dn.net/?f=p)
and
![q](https://tex.z-dn.net/?f=q)
supposed to be? Can't answer a/b without that information.
I'll come back to part (c) in a moment. If we can show
![\mathbf f](https://tex.z-dn.net/?f=%5Cmathbf%20f)
is conservative, this part will be a breeze.
For part (d), to show whether
![\mathbf f](https://tex.z-dn.net/?f=%5Cmathbf%20f)
is conservative, we have to show that there is a scalar function
![f](https://tex.z-dn.net/?f=f)
such that
![\nabla f=\mathbf f](https://tex.z-dn.net/?f=%5Cnabla%20f%3D%5Cmathbf%20f)
. It appears that you've written
![\mathbf f=-y\,\mathbf i+x\,\mathbf j](https://tex.z-dn.net/?f=%5Cmathbf%20f%3D-y%5C%2C%5Cmathbf%20i%2Bx%5C%2C%5Cmathbf%20j)
I'm not sure what to make of the
![x^2+y^2](https://tex.z-dn.net/?f=x%5E2%2By%5E2)
that follows.
![\nabla f=\mathbf f](https://tex.z-dn.net/?f=%5Cnabla%20f%3D%5Cmathbf%20f)
means that
![\dfrac{\partial f}{\partial x}=-y](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20x%7D%3D-y)
![\dfrac{\partial f}{\partial y}=x](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20y%7D%3Dx)
Integrating the first PDE with respect to
![x](https://tex.z-dn.net/?f=x)
gives
![f(x,y)=-xy+g(y)](https://tex.z-dn.net/?f=f%28x%2Cy%29%3D-xy%2Bg%28y%29)
Differentiating with respect to
![y](https://tex.z-dn.net/?f=y)
gives
![\dfrac{\partial f}{\partial y}=-x+\dfrac{\mathrm dg}{\mathrm y}=x](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20y%7D%3D-x%2B%5Cdfrac%7B%5Cmathrm%20dg%7D%7B%5Cmathrm%20y%7D%3Dx)
![\implies\dfrac{\mathrm dg}{\mathrm dy}=2x](https://tex.z-dn.net/?f=%5Cimplies%5Cdfrac%7B%5Cmathrm%20dg%7D%7B%5Cmathrm%20dy%7D%3D2x)
But we assumed that
![g(y)](https://tex.z-dn.net/?f=g%28y%29)
is a function of
![y](https://tex.z-dn.net/?f=y)
alone, which means there is no solution for
![g](https://tex.z-dn.net/?f=g)
, and therefore no solution for
![f](https://tex.z-dn.net/?f=f)
. Hence
![f](https://tex.z-dn.net/?f=f)
is not conservative.
Back to part (c).
![\mathbf f](https://tex.z-dn.net/?f=%5Cmathbf%20f)
is not conservative, so we have to compute the line integral the "long" way. Replacing
![x=\cos t](https://tex.z-dn.net/?f=x%3D%5Ccos%20t)
and
![y=\sin t](https://tex.z-dn.net/?f=y%3D%5Csin%20t)
, we have
![\displaystyle\int_{\mathcal C}\mathbf f\cdot\mathrm d\mathbf r=\int_{t=0}^{t=2\pi}(-\sin t\,\mathbf i+\cos t\,\mathbf j)\cdot(-\sin t\,\mathbf i+\cos t\,\mathbf j)\,\mathrm dt](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_%7B%5Cmathcal%20C%7D%5Cmathbf%20f%5Ccdot%5Cmathrm%20d%5Cmathbf%20r%3D%5Cint_%7Bt%3D0%7D%5E%7Bt%3D2%5Cpi%7D%28-%5Csin%20t%5C%2C%5Cmathbf%20i%2B%5Ccos%20t%5C%2C%5Cmathbf%20j%29%5Ccdot%28-%5Csin%20t%5C%2C%5Cmathbf%20i%2B%5Ccos%20t%5C%2C%5Cmathbf%20j%29%5C%2C%5Cmathrm%20dt)
Answer:
-1/4
Step-by-step explanation:
Perpendicular means negative reciprocal of the current slope.
Answer:
Across
5 across: percent of change
6. Ratio
7. Decrease
Down.
1. Percent
2 interest
3 proportion
4 sales tax
Step-by-step explanation:
Answer:
![\frac{26}{100}](https://tex.z-dn.net/?f=%5Cfrac%7B26%7D%7B100%7D)
Step-by-step explanation:
can stay untouched, but
has to be changed to
. Now that they have they have the same denominator, you can add them.
, which can be simplified, which is
.