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For a constant \(c\) in spherical coordinates \( (\rho,\theta,\phi)\), find the shape described by the equation \(\phi = c\)

(A) Line
(B) Circle
(C) Plane
(D) Sphere
(E) Cylinder
(F) Cone

 Apr 12, 2020
 #1
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Because of the spherical coordinates, phi = c traces a cylinder.  So the answer is (E) cylinder.

 Apr 13, 2020
 #2
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In spherical coordinates, the \(\phi\) represents the angle between (the line segment between the origin and a point) and the x-y plane.

If \(\phi = c\), that means the angle \(\phi\) is constant. Construct a line L such that the angle between L and the x-y plane is \(\phi\), then revolve it around the z-axis. For any line segment Lin the resulting figure, the angle between L0 and x-y plane is \(\phi\). Therefore the resulting figure is a cone.

 

Therefore the answer is (F).

 Apr 14, 2020

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