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?

The required, there is no part of the sphere x² + y² + z² = 16 that lies above the cone z = x² + y², where z > x² + y².

To find the part of the sphere x² + y² + z² = 16 that lies above the cone z = x² + y², where z > x² + y², we can use spherical coordinates. In spherical coordinates, the equations for the sphere and the cone are simpler.

Spherical coordinates are represented as (ρ, θ, φ), where ρ is the radial distance, θ is the azimuthal angle (measured from the positive x-axis in the xy-plane), and φ is the polar angle (measured from the positive z-axis).

For the sphere x² + y² + z² = 16, the spherical representation is:

ρ = 4 (since ρ² = x² + y² + z² = 16)

For the cone z = x² + y², the spherical representation is:

ρ = ρ (since ρ^2 = x² + y²)

Now, to find the part of the sphere that lies above the cone (z > x² + y^2), we need to restrict the values of φ.

When z > x² + y², we have z = ρ cos(φ) > ρ².

Since ρ = 4, we get 4 cos(φ) > 4², which simplifies to cos(φ) > 4.

However, the range of φ in spherical coordinates is 0 ≤ φ ≤ π, which means that the values of φ that satisfy cos(φ) > 4 are not within the valid range.

Therefore, there is no part of the sphere x² + y² + z² = 16 that lies above the cone z = x² + y², where z > x² + y².

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Final answer:

We use the given equations of the sphere and cone and express them in spherical coordinates. The sphere lies on or above the cone when z's value in the sphere equation is greater or equal than z's value in the cone equation. One method is to use spherical coordinates and represent the radius and polar angle in terms of u and v.

Explanation:

The question involves spherical and rectangular coordinates and the relationship between the two. We are given the sphere's equation as x^2 + y^2 + z^2 = 16 and the cone's equation as z = x^2 + y^2. Here's one way to think of the part of the sphere that lies on or above the cone. If we view z=x^2 + y^2 as a function of x and y, the sphere lies above this cone when z's value in the equation of the sphere is greater or equal to the value of z in the cone's equation. To express x, y, and z in terms of u and/or v, you can use a method such as spherical coordinates.

In spherical coordinates, the relationship between spherical and rectangular coordinates can be represented as:

• x = r sin θ cos φ
• y = r sin θ sin φ
• z = r cos θ

Here r, θ, and φ are the radius, polar, and azimuthal angles respectively, which we can let u and v represent. One potential assignment is to let r=u and θ=v, assuming we want only two parameters.

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