Calculating Surface Area of a Cone with a Vertex in the XY-Plane

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Discussion Overview

The discussion revolves around calculating the surface area of a cone defined by the equation z=2√(x²+y²) and its relationship to a specified area in the xy-plane. The context includes aspects of calculus, particularly multiple integration, and the interpretation of the problem statement.

Discussion Character

  • Homework-related, Mathematical reasoning, Conceptual clarification

Main Points Raised

  • Mike seeks assistance with a homework problem regarding the surface area of a cone.
  • Some participants question whether the problem is asking for the area of the cone's base or the slant area of the cone.
  • There is a suggestion that the problem may involve determining the height of the cone at which the base has an area of 5, with a formula provided for the radius based on the height.

Areas of Agreement / Disagreement

Participants do not appear to agree on the interpretation of the problem, with multiple competing views regarding whether the focus is on the base area or the slant area of the cone.

Contextual Notes

There is ambiguity in the problem statement regarding the definitions of "area" and whether it refers to the base or the slant surface of the cone. The relationship between the height of the cone and the area of the base is also not fully resolved.

JaysFan31
Find the surface area of the cone z=2sqrt(x^2+y^2) and above a region in the xy-plane with area 5.

If anyone could help me with this problem, I would really appreciate it.
Thanks.

Mike
 
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Do you mean to find the area of the cone of equation z=2sqrt(x^2+y^2) whose base has area 5 ?
 
No. Not sure what it means. Just one of my Calculus III Multiple Integration homework problems word for word and I have no idea.

Mike
 
quasar987's point is that the cone [itex]z= 2\sqrt{x^2+ y^2}[/itex] has its vertex in the xy-plane, not a base. Perhaps you mean, as he suggested, the slant area of that cone up to the point the base would have an area of 5. (Obviously, the radius of the base of the cone is [itex]\frac{\sqrt{z}}{2}[/itex]. What value of z gives area 5?)
 

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