Answer:
Buzz and woody should see a doctor because they are ugly
Step-by-step explanation:
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![D](https://tex.z-dn.net/?f=D)
is a right triangle with base length 1 and height 8, so the area of
![D](https://tex.z-dn.net/?f=D)
is
![\dfrac12(1)(8)=4](https://tex.z-dn.net/?f=%5Cdfrac12%281%29%288%29%3D4)
.
The average value of
![f(x,y)](https://tex.z-dn.net/?f=f%28x%2Cy%29)
over
![D](https://tex.z-dn.net/?f=D)
is given by the ratio
![\dfrac{\displaystyle\iint_Df(x,y)\,\mathrm dA}{\displaystyle\iint_D\mathrm dA}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cdisplaystyle%5Ciint_Df%28x%2Cy%29%5C%2C%5Cmathrm%20dA%7D%7B%5Cdisplaystyle%5Ciint_D%5Cmathrm%20dA%7D)
The denominator is just the area of
![D](https://tex.z-dn.net/?f=D)
, which we already know. The average value is then simplified to
![\displaystyle\frac74\iint_Dxy\,\mathrm dA](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cfrac74%5Ciint_Dxy%5C%2C%5Cmathrm%20dA)
In the
![x,y](https://tex.z-dn.net/?f=x%2Cy)
-plane, we can describe the region
![D](https://tex.z-dn.net/?f=D)
as all points
![(x,y)](https://tex.z-dn.net/?f=%28x%2Cy%29)
that lie between the lines
![y=0](https://tex.z-dn.net/?f=y%3D0)
and
![y=8x](https://tex.z-dn.net/?f=y%3D8x)
(the lines which coincide with the triangle's base and hypotenuse, respectively), taking
![0\le x\le1](https://tex.z-dn.net/?f=0%5Cle%20x%5Cle1)
. So, the integral is given by, and evaluates to,
![\displaystyle\frac74\int_{x=0}^{x=1}\int_{y=0}^{y=8x}xy\,\mathrm dy\,\mathrm dx=\frac78\int_{x=0}^{x=1}xy^2\bigg|_{y=0}^{y=8x}\,\mathrm dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cfrac74%5Cint_%7Bx%3D0%7D%5E%7Bx%3D1%7D%5Cint_%7By%3D0%7D%5E%7By%3D8x%7Dxy%5C%2C%5Cmathrm%20dy%5C%2C%5Cmathrm%20dx%3D%5Cfrac78%5Cint_%7Bx%3D0%7D%5E%7Bx%3D1%7Dxy%5E2%5Cbigg%7C_%7By%3D0%7D%5E%7By%3D8x%7D%5C%2C%5Cmathrm%20dx)
![=\displaystyle56\int_{x=0}^{x=1}x^3\,\mathrm dx](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle56%5Cint_%7Bx%3D0%7D%5E%7Bx%3D1%7Dx%5E3%5C%2C%5Cmathrm%20dx)
![=14x^4\bigg|_{x=0}^{x=1}](https://tex.z-dn.net/?f=%3D14x%5E4%5Cbigg%7C_%7Bx%3D0%7D%5E%7Bx%3D1%7D)
Hope you are able to understand the solution. :-D
No, because the 9 pounds would cost 22.5, and that is 5 cents more than he has
X= number of tubs Kasey sold
3x= number tubs sister sold
Add the number sold by Kasey and her sister to equal the total sold (88).
x + 3x= 88
combine like terms
4x= 88
divide both sides by 4
x= 22 number tubs Kasey sold
Number Sold by Sister
3x= 3(22)= 66
CHECK:
x + 3x= 88
22 + 3(22)= 88
22 + 66= 88
88= 88
ANSWER: Kasey sold 22 tubs and her sister sold 66 tubs.
Hope this helps! :)