Answer:
10
Step-by-step explanation:
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![y = {(x + 1)}^{2} - 1](https://tex.z-dn.net/?f=y%20%3D%20%7B%28x%20%2B%201%29%7D%5E%7B2%7D%20-%201)
The one is a vertical translation of the parent function to the left of one
![f(x) = {x}^{2}](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%7Bx%7D%5E%7B2%7D%20)
making it -1.
Answer: 140 Hot dogs and 83 sodas.
Step-by-step explanation:
Alright, so we will say that h = hot dogs and s = sodas.
h + s = 223
I have used the guess and test strategy to get:
h = 140
140 - 57 = 83
s = 83
Check: 83 + 140 = 223
We can parameterize this part of a cone by
![\mathbf s(u,v)=\left\langle u\cos v,u\sin v,\dfrac hau\right\rangle](https://tex.z-dn.net/?f=%5Cmathbf%20s%28u%2Cv%29%3D%5Cleft%5Clangle%20u%5Ccos%20v%2Cu%5Csin%20v%2C%5Cdfrac%20hau%5Cright%5Crangle)
with
![0\le u\le a](https://tex.z-dn.net/?f=0%5Cle%20u%5Cle%20a)
and
![0\le v\le2\pi](https://tex.z-dn.net/?f=0%5Cle%20v%5Cle2%5Cpi)
. Then
![\mathrm dS=\|\mathbf s_u\times\mathbf s_v\|\,\mathrm du\,\mathrm dv=\sqrt{1+\dfrac{h^2}{a^2}}u\,\mathrm du\,\mathrm dv](https://tex.z-dn.net/?f=%5Cmathrm%20dS%3D%5C%7C%5Cmathbf%20s_u%5Ctimes%5Cmathbf%20s_v%5C%7C%5C%2C%5Cmathrm%20du%5C%2C%5Cmathrm%20dv%3D%5Csqrt%7B1%2B%5Cdfrac%7Bh%5E2%7D%7Ba%5E2%7D%7Du%5C%2C%5Cmathrm%20du%5C%2C%5Cmathrm%20dv)
The area of this surface (call it
![\mathcal S](https://tex.z-dn.net/?f=%5Cmathcal%20S)
) is then