A solid lies above the cone z = x2 + y2 and below the sphere x2 + y2 + z2 = z. write a description of the solid in terms of inequalities involving spherical coordinates

Question

Nola Boone

Mathematics

6 May, 08:18

A solid lies above the cone z = x2 + y2 and below the sphere x2 + y2 + z2 = z. write a description of the solid in terms of inequalities involving spherical coordinates

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Austin Alexander · Answer 1

Since solid is above the cone z > = √x^2 + y^2 or z^2 > = x^2 + y^2 or 2z^2 > = x^2+z^2 = p^2 or 2p^2 cos^2 φ > = p^2. Thus cos φ > = 1/√2 since the cone upwards. Thus 0 < = φ < = pi / 4.

On the other hand in spherical coordinates the sphere here is pcos φ = p^2 so 0 < = cos φ since the solid lies between the sphere. Thus, making it the two inequalities.