#Multivariable Calc Help Needed

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dawn musk
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your level surface at a point C(a,b,c) is g(x,y,z)=g(a,b,c)

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you have C(100,0,0) as your point

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and you've found that g(100,0,0) is 30

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so g(x,y,z)=30

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sub in your g(x,y,z) and you've got your level surface

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10000=x²+y²+z²
so you have a sphere of radius 100

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@novel needle

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you want the level surface that passes through the point that you just calculated your g for

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it cut out at g=30 so you make g(x,y,z)=30

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then you simplify

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our 100²=x²+y²+z² is our surface

dawn musk
novel needle
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but

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how do we know that the given equation is of a sphere form?

dawn musk
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because the general form of a sphere is r²=(x-a)²+(y-b)²+(z-c)² with origin (a,b,c)

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we just have origin (0,0,0)

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and r 100

novel needle
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that's wut i'm confused about

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as to how we know

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the origin is (0,0,0)

dawn musk
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because that's what your formula for g indicates

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you'd assume your location (x,y,z) is relative to some relevant object, hence it makes sense that it's relative to the presumable centre of your mass

novel needle
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So the shape of our level surface is indeed a sphere?

dawn musk
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yes

novel needle
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and then lastly, how would u follow up with this:

dawn musk
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you know your positional coordinates are given in km and the radius of your sphere is 100, so you just use the surface area formula for a sphere

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40,000π sqkm

novel needle
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hmm

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where did u get 400pi from?

dawn musk
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nvm I thought that was a bit low

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40,000π sqkm

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4πr²

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surface area of a sphere

novel needle
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So,

Since we know that the shape our level surface is a sphere,

We can use the formula for the surface area of a sphere which is SA = 4pir^2

Since the equation for our surface level is 10000 = x^2 + y^2 + z^2

We know that r = 100 as 100^2 = 10000

Therefore, the surface area of the level surface in sqaure kilometers is 40,000pi sqkm

dawn musk
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yes

novel needle
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That reasoning makes sense and adds up right?

dawn musk
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yup

novel needle
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wow

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You truly are

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A legendary legend

dawn musk
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no problem

novel needle
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Rlly appreciate the help sir. Thank you so much!

dawn musk
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you're welcome, close the thread when you're ready

novel needle
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will, do. just reading it once more

novel needle
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+close