Question

Find a parametric representation for the part of the sphere (a surface, not a solid) of radius 4 centered at the origin that lies (a) inside the cone z= 3(x2 +y 2) (b) inside the cone x= 31(y 2 +z 2 )

c) inside the cone y= x +z 2
Hint: you may want to use spherical-like coordinates, where the roles of x,y and z are permuted.

Answer 1

A parametric representation for the part of the sphere of radius 4 centered at the origin that lies inside the given cones can be obtained using spherical coordinates.

Step-by-step explanation:

To find a parametric representation for the part of the sphere of radius 4 centered at the origin that lies inside the given cones, we can use spherical coordinates. Let's start with the first cone, z = 3(x^2 + y^2). Rearranging the equation, we have x^2 + y^2 = z/3. Since we want the part of the sphere that lies inside this cone, we can set the range for the spherical coordinates as r ∈ [0, 4], θ ∈ [0, 2π], and φ ∈ [0, π/2]. The parametric representation is then x = r sin(φ) cos(θ), y = r sin(φ) sin(θ), and z = r cos(φ), where r, θ, and φ are the spherical coordinates.

We can follow a similar approach for the second and third cones. For the second cone, x = 31(y^2 + z^2), we can set the range for the spherical coordinates as r ∈ [0, 4], θ ∈ [0, π/2], and φ ∈ [0, 2π]. The parametric representation is x = r cos(θ), y = r sin(θ) sin(φ), and z = r sin(θ) cos(φ). For the third cone, y = x + z^2, we can set the range for the spherical coordinates as r ∈ [0, 4], θ ∈ [0, π/2], and φ ∈ [0, π]. The parametric representation is x = r sin(θ) sin(φ), y = r sin(θ) cos(φ), and z = r cos(θ), where r, θ, and φ are the spherical coordinates.

Answer 2

Read carefully below

Step-by-step explanation:

We can use spherical coordinates to represent the points on the sphere. These coordinates are typically denoted as (r, θ, φ), where r is the radius of the sphere, θ is the polar angle, and φ is the azimuthal angle.

In order to find a parametric representation of the part of the sphere that lies inside the given cones, we can use the following approach:

First, we need to find the intersection of the sphere and the cone. This is the set of points that satisfy the equations of both the sphere and the cone.

Then, we can use spherical coordinates to represent these points on the sphere.

Finally, we can use these spherical coordinates as parameters in a parametric equation of the form x = x(r, θ, φ), y = y(r, θ, φ), z = z(r, θ, φ) to represent the points on the part of the sphere that lies inside the cone.

Let's use this approach to find a parametric representation of the part of the sphere that lies inside the cone z = 3(x^2 + y^2) in the first case.

To find the intersection of the sphere and the cone, we need to solve the following system of equations:

x^2 + y^2 + z^2 = 16 (equation of the sphere)

z = 3(x^2 + y^2) (equation of the cone)

Substituting the second equation into the first equation, we get:

x^2 + y^2 + 9(x^4 + y^4) = 16

This equation represents an ellipse in the xy-plane. We can use the parametric equations of an ellipse to represent the points on this ellipse:

x = a * cos(t)

y = b * sin(t)

where a and b are the semi-major and semi-minor axes of the ellipse, respectively, and t is a parameter that varies from 0 to 2π.

We can now use spherical coordinates to represent the points on the part of the sphere that lies inside the cone. We can let r be the radius of the sphere (r = 4), θ be the polar angle (0 ≤ θ ≤ π), and φ be the azimuthal angle (0 ≤ φ ≤ 2π).

The parametric equations of the sphere in spherical coordinates are:

x = r * sin(θ) * cos(φ)

y = r * sin(θ) * sin(φ)

z = r * cos(θ)

Substituting the equations for x and y from the ellipse into the equations for x and y from the sphere, we get:

x = 4 * sin(θ) * cos(φ) = a * cos(t)

y = 4 * sin(θ) * sin(φ) = b * sin(t)

z = 4 * cos(θ) = 3(x^2 + y^2) = 3(a^2 * cos^2(t) + b^2 * sin^2(t))

We can solve for a and b in terms of r and θ:

a = 4 * sin(θ)

b = 4 * sin(θ)

Therefore, the parametric equations for the part of the sphere that lies inside the cone z