The part of the sphere x2 + y2 + z2 = 16 that lies above the cone z = x2 + y2 . (enter your answer as a comma-separated list of equations. let x, y, and z be in terms of u and/or v.) where z > x2 + y2?

(x,y,z)=(ucos(v), usin(v), )

where and

Step-by-step explanation:

Equation of a cone is

Equation of a paraboloid is

I have parametrised cone here. Please note that equation for cone in the question, is actually a paraboloid.

Imagine a sphere of radius 4, centered at origin and intersecting a cone also centered at origin and height along positive z-axis, given by the equations

where

Solving for these two equations, and substituting for z in the equation of sphere, we get a circle of radius units.

The equation of intersecting circle is:

Now, according to question, parametrizing this region of circle using parameters u and v.

Consider cylindrical co-ordinates: (r,θ,z)

In cylindrical co-ordinates

(x, y, z)= (r cos(θ),  r sin(θ), z)

Eliminating z, and changing (r, θ)=(u,v)

For cone: x=ucos(v)

y= usin(v)

z=

or (x,y,z)=(ucos(v), usin(v), )

where and

where are the options?

step-by-step explanation:

because they could drop even lower..