Question

Convert to a Rectangular equation. Match each Item (i)-(iv) to an Item (A)-(J). A circle centered at origin with radius 3 (F) Half line x=y=0 and z≤0

Answer 1

The conversion of a circle centered at the origin with a radius of 3 to a rectangular equation is x² + y² = 9. The half-line represented by x = y = 0 and z ≤ 0 is already in Cartesian form.

Step-by-step explanation:

The student is asking for the conversion of a polar or cylindrical equation to a rectangular equation. Specifically, they want to convert the description of a circle centered at the origin with a radius of 3 and a half-line where x and y are 0 and z is less than or equal to 0, to the appropriate rectangular (also known as Cartesian) coordinates.

The equation of a circle centered at the origin in Cartesian coordinates is x² + y² = r², where r is the radius. Thus, for a circle with radius 3, the equation would be x² + y² = 3², which simplifies to x² + y² = 9. As for the half-line, since it lies along the z-axis and includes all points where z is less than or equal to 0, it is already denoted in Cartesian coordinates by the inequalities x = 0, y = 0, and z ≤ 0