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anonymous

  • 5 years ago

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?

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  1. JamesJ
    • 5 years ago
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    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?

  2. anonymous
    • 5 years ago
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    I still don't understand...

  3. anonymous
    • 5 years ago
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    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|??

  4. JamesJ
    • 5 years ago
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    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.

  5. JamesJ
    • 5 years ago
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    Therefore is your shape consistent with a cone or a paraboloid?

  6. anonymous
    • 5 years ago
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    It would have to be more consistent with a cone.

  7. JamesJ
    • 5 years ago
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    Yes, because z = |x| is a straight line for x > 0 and a different straight line for x < 0

  8. JamesJ
    • 5 years ago
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    and a cone has straight edges along the x-axis, while a paraboloid does not.

  9. anonymous
    • 5 years ago
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    Thanks James! I don't know how you're so boss at calculus, but I really appreciate your help and detailed explanations :)

  10. JamesJ
    • 5 years ago
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    I'm a fallen angel. Had a few millennia practicing conic sections.

  11. anonymous
    • 5 years ago
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    Are you a professor???

  12. JamesJ
    • 5 years ago
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    In a parallel universe.

  13. anonymous
    • 5 years ago
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    I love you man. All I have to say :).

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