First find the critical points of <em>f</em> :
![f(x,y)=2x^2+3y^2-4x-5=2(x-1)^2+3y^2-7](https://tex.z-dn.net/?f=f%28x%2Cy%29%3D2x%5E2%2B3y%5E2-4x-5%3D2%28x-1%29%5E2%2B3y%5E2-7)
![\dfrac{\partial f}{\partial x}=2(x-1)=0\implies x=1](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20x%7D%3D2%28x-1%29%3D0%5Cimplies%20x%3D1)
![\dfrac{\partial f}{\partial y}=6y=0\implies y=0](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20y%7D%3D6y%3D0%5Cimplies%20y%3D0)
so the point (1, 0) is the only critical point, at which we have
![f(1,0)=-7](https://tex.z-dn.net/?f=f%281%2C0%29%3D-7)
Next check for critical points along the boundary, which can be found by converting to polar coordinates:
![f(x,y)=f(10\cos t,10\sin t)=g(t)=295-40\cos t-100\cos^2t](https://tex.z-dn.net/?f=f%28x%2Cy%29%3Df%2810%5Ccos%20t%2C10%5Csin%20t%29%3Dg%28t%29%3D295-40%5Ccos%20t-100%5Ccos%5E2t)
Find the critical points of <em>g</em> :
![\dfrac{\mathrm dg}{\mathrm dt}=40\sin t+200\sin t\cos t=40\sin t(1+5\cos t)=0](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dg%7D%7B%5Cmathrm%20dt%7D%3D40%5Csin%20t%2B200%5Csin%20t%5Ccos%20t%3D40%5Csin%20t%281%2B5%5Ccos%20t%29%3D0)
![\implies\sin t=0\text{ OR }1+5\cos t=0](https://tex.z-dn.net/?f=%5Cimplies%5Csin%20t%3D0%5Ctext%7B%20OR%20%7D1%2B5%5Ccos%20t%3D0)
![\implies t=n\pi\text{ OR }t=\cos^{-1}\left(-\dfrac15\right)+2n\pi\text{ OR }t=-\cos^{-1}\left(-\dfrac15\right)+2n\pi](https://tex.z-dn.net/?f=%5Cimplies%20t%3Dn%5Cpi%5Ctext%7B%20OR%20%7Dt%3D%5Ccos%5E%7B-1%7D%5Cleft%28-%5Cdfrac15%5Cright%29%2B2n%5Cpi%5Ctext%7B%20OR%20%7Dt%3D-%5Ccos%5E%7B-1%7D%5Cleft%28-%5Cdfrac15%5Cright%29%2B2n%5Cpi)
where <em>n</em> is any integer. We get 4 critical points in the interval [0, 2π) at
![t=0\implies f(10,0)=155](https://tex.z-dn.net/?f=t%3D0%5Cimplies%20f%2810%2C0%29%3D155)
![t=\cos^{-1}\left(-\dfrac15\right)\implies f(-2,4\sqrt6)=299](https://tex.z-dn.net/?f=t%3D%5Ccos%5E%7B-1%7D%5Cleft%28-%5Cdfrac15%5Cright%29%5Cimplies%20f%28-2%2C4%5Csqrt6%29%3D299)
![t=\pi\implies f(-10,0)=235](https://tex.z-dn.net/?f=t%3D%5Cpi%5Cimplies%20f%28-10%2C0%29%3D235)
![t=2\pi-\cos^{-1}\left(-\dfrac15\right)\implies f(-2,-4\sqrt6)=299](https://tex.z-dn.net/?f=t%3D2%5Cpi-%5Ccos%5E%7B-1%7D%5Cleft%28-%5Cdfrac15%5Cright%29%5Cimplies%20f%28-2%2C-4%5Csqrt6%29%3D299)
So <em>f</em> has a minimum of -7 and a maximum of 299.
The question is asking to convert the said decimal value in a simplest fraction form, base on my research and further calculation, I would say that the answer would be 3/10. I hope you are satisfied with my answer and feel free to ask for more
Answer:
$13.77 per person in the family
Step-by-step explanation:
Cost of the meal = $54.65
Sales tax rate = 8%
Sales tax amount = 8% of $54.65
= 8/100 × 54.65
= 0.08 × 54.65
= $4.372
Tip = 18% of $54.65
= 18/100 × 54.65
= 0.18 × 54.65
= $9.837
Total cost of the dinner = Cost of the meal + Sales tax amount + Tip
= $54.65 + $4.372 + $9.837
= $68.859
There are 5 people in the Holcombe family
What was the cost per person for the meal, including sales tax and tip?
cost per person for the meal, including sales tax and tip = Total cost of the dinner / total number of people in the family
= $68.859 / 5
= $13.7718
Approximately
$13.77 per person in the family
Answer:
The correct option is B.
Step-by-step explanation:
The given equation is
..... (1)
Where, y is distance from home for an ant t minutes after starting out from a location near its home.
The slope intercept form of a line is
.... (2)
Where, m is slope and b is y-intercept or initial value.
On comparing (1) and (2), we get
![m=2, b=5](https://tex.z-dn.net/?f=m%3D2%2C%20b%3D5)
Here, 5 represent the initial distance of ant from its home.
It means the ant is moving at 2 feet per minute and the ant started out 5 feet from its home.
Therefore option B is correct.