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# Spherical Coordinate Precalculus Problem

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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

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Because of the spherical coordinates, phi = c traces a cylinder.  So the answer is (E) cylinder.

Apr 13, 2020
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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.