# Give examples of equations for the following common surfaces:

1. Plane:

- Equation: $a x+b y+c z=d$
- Easiest Coordinate System: Cartesian coordinates, because the plane can be easily described with a linear equation involving $x, y$, and $z$.

2. Sphere:

- Equation: $x^{2}+y^{2}+z^{2}=r^{2}$
- Easiest Coordinate System: Spherical coordinates, because the radius and angles naturally describe the spherical shape.

3. Elliptic Paraboloid:

- Equation: $z=\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}$
- Easiest Coordinate System: Cartesian coordinates, since it provides a straightforward way to represent the quadratic terms.

4. Hyperbolic Paraboloid:

- Equation: $z=\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}$
- Easiest Coordinate System: Cartesian coordinates, for the same reasons as the elliptic paraboloid.

5. Circular Cylinder:

- Equation: $x^{2}+y^{2}=r^{2}$
- Easiest Coordinate System: Cylindrical coordinates, because the equation involves a fixed radius from a central axis.

6. Half Cone:

- Equation: $z^{2}=x^{2}+y^{2}$
- Easiest Coordinate System: Cylindrical coordinates, since it can naturally describe the conical shape using a linear relationship between height and radial distance.

Question

# Give examples of equations for the following common surfaces:

1. Plane:

- Equation: $a x+b y+c z=d$
- Easiest Coordinate System: Cartesian coordinates, because the plane can be easily described with a linear equation involving $x, y$, and $z$.

2. Sphere:

- Equation: $x^{2}+y^{2}+z^{2}=r^{2}$
- Easiest Coordinate System: Spherical coordinates, because the radius and angles naturally describe the spherical shape.

3. Elliptic Paraboloid:

- Equation: $z=\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}$
- Easiest Coordinate System: Cartesian coordinates, since it provides a straightforward way to represent the quadratic terms.

4. Hyperbolic Paraboloid:

- Equation: $z=\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}$
- Easiest Coordinate System: Cartesian coordinates, for the same reasons as the elliptic paraboloid.

5. Circular Cylinder:

- Equation: $x^{2}+y^{2}=r^{2}$
- Easiest Coordinate System: Cylindrical coordinates, because the equation involves a fixed radius from a central axis.

6. Half Cone:

- Equation: $z^{2}=x^{2}+y^{2}$
- Easiest Coordinate System: Cylindrical coordinates, since it can naturally describe the conical shape using a linear relationship between height and radial distance.

StudyXY · Accepted Answer

I'll solve this problem by providing the equations and explanations for each surface as shown in the image. : Plane : Sphere : Elliptic Paraboloid : Hyperbolic Paraboloid : Circular Cylinder : Half Cone The provided equations represent six fundamental geometric surfaces, each described using specific coordinate systems that best capture their geometric properties.

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Step 1
I'll solve this problem by providing the equations and explanations for each surface as shown in the image.

Step 2
: Plane

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