Mathematics: What is if g(x,y,z) = x + y and s is the first octant portion of the plane 2x + 3y + z = 6 ?

What is if g(x,y,z) = x + y and s is the first

Ответ:

neylabaker7489

05.01.2022

The question asks for the value of $I=\int\int_Sx+y\textrm{ }dS$ where $S=\{(x,y,z)\mid2x+3z+y=6,x\ge0,y\ge0,z\ge0\}$ .

First let's look at what that surface looks like.

Letting y=z=0

yields

Letting

yields

Letting

yields

Therefore

is the area of the triangle defined by the three points (3,0,0),(0,2,0),(0,0,6)

.

We can thus reformulate the integral as $I=\int_{z=0}^6\int_{x=0}^{6-z}x+ydxdz$ .

By definition on the plane $y=\dfrac{6-2x-z}3$ thus $I=\int_{z=0}^6\int_{x=0}^{6-z}x+\frac{6-2x-z}3dxdz=\int_{z=0}^6\int_{x=0}^{6-z}2+\frac x3-\frac z3 dxdz$

$I=\int_{z=0}^6\left[2x+\frac{x^2}6-\frac{zx}3\right]_{x=0}^{6-z}dz=\int_{z=0}^62(6-z)+\frac{(6-z)^2}6-\frac{z(6-z)}3\right]dz$

$I=\int_{z=0}^6\frac{z^2}2-6z+18=\left[\frac{z^ 3}6-3z^2+18z\right]_{z=0}^6=36-108+108$

Hence $\boxed{I=\int\int_Sx+y\textrm{ }dS=36}$

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Ответ:

alysiareyes

28.01.2023

Mathematics

x = amount of children attended the event

y = amount of adults attended the event

they want to raise $1000, and the organizing costs $200, so the revenue, incoming capital, must be 1000 + 200, so the cost is covered and they're left with the $1000 they're after.

since is $6 per child, and "x" children went to it, they total amount of revenue from the ticket sales for all children is 6*x, or 6x.

since is $8 per adult, and "y" adults went to it, they total amount of revenue from the ticket sales for all adults is 8*y, or 8y.

6x + 8y = 1200.

if the event were attended by only adults, then 8y = 1200.

$\bf 8y=1200\implies y=\cfrac{1200}{8}\implies y=150$

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What is if g(x,y,z) = x + y and s is the first octant portion of the plane 2x + 3y + z = 6 ?

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