anonymous
  • anonymous
Determine the type of solid 0 <= theta <= pi/2; r <= z <= 7 Would this be a quarter of a cone or a quarter of a paraboloid?
Mathematics
chestercat
  • chestercat
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JamesJ
  • JamesJ
Think about what happens along the x-axis. There \[ r =\sqrt{x^2 + y^2} = \sqrt{x^2 + 0^2} = |x| \] Hence if \( z = |x| \), is that consistent with a cone or a paraboloid?
anonymous
  • anonymous
I still don't understand...
anonymous
  • anonymous
So would this be looking like a cone? Since parabolas are curvy, which doesn't closely represent |x|? Whereas, a cone on a 2D graph would look like |x|??

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JamesJ
  • JamesJ
You have that r <= z <= 7 Hence |x| <= z <= 7 Now at the boundary of the object, r = z, or along the x-axis z = r = |x|. If the object were a cone, would we expect the cross section along the x-axis to be ... what? A straight line, a parabola, what? If the object were a paraboloid, what would you expect of the cross-section? A straight line, parabola or something else? Given that z = |x|, is that straight line, parabola or something else? Therefore is your shape consistent with a cone or a paraboloid.
JamesJ
  • JamesJ
Therefore is your shape consistent with a cone or a paraboloid?
anonymous
  • anonymous
It would have to be more consistent with a cone.
JamesJ
  • JamesJ
Yes, because z = |x| is a straight line for x > 0 and a different straight line for x < 0
JamesJ
  • JamesJ
and a cone has straight edges along the x-axis, while a paraboloid does not.
anonymous
  • anonymous
Thanks James! I don't know how you're so boss at calculus, but I really appreciate your help and detailed explanations :)
JamesJ
  • JamesJ
I'm a fallen angel. Had a few millennia practicing conic sections.
anonymous
  • anonymous
Are you a professor???
JamesJ
  • JamesJ
In a parallel universe.
anonymous
  • anonymous
I love you man. All I have to say :).

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