15–22. Sets in polar coordinates Sketch the following sets of points.

2 ≤ r ≤ 8

검증된 단계별 안내

Understand that the problem asks to sketch the set of points in polar coordinates where the radius \(r\) satisfies \(2 \leq r \leq 8\).

Recall that in polar coordinates, a point is represented as \((r, \theta)\), where \(r\) is the distance from the origin and \(\theta\) is the angle from the positive x-axis.

Since there is no restriction on \(\theta\), it means \(\theta\) can take any value from \(0\) to \(2\pi\) (or all angles around the origin).

The inequality \(2 \leq r \leq 8\) describes all points whose distance from the origin is at least 2 units and at most 8 units, forming a ring-shaped region (an annulus) between two circles of radii 2 and 8.

To sketch, draw two concentric circles centered at the origin: one with radius 2 and another with radius 8. The region between these two circles represents the set of points satisfying \(2 \leq r \leq 8\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Polar Coordinates System

The polar coordinate system represents points in a plane using a radius and an angle, denoted as (r, θ). Here, r is the distance from the origin, and θ is the angle measured from the positive x-axis. Understanding this system is essential for interpreting and sketching regions defined by inequalities involving r and θ.

Inequalities in Polar Coordinates

Inequalities like 2 ≤ r ≤ 8 describe regions in the plane where the radius r lies between two values. This defines an annular region (ring-shaped area) between two circles centered at the origin with radii 2 and 8. Recognizing how inequalities restrict r and θ helps in visualizing and sketching the corresponding sets.

Graphing Regions in Polar Coordinates

Sketching sets in polar coordinates involves plotting all points that satisfy given conditions on r and θ. For 2 ≤ r ≤ 8, the graph is the area between two concentric circles. Understanding how to translate these inequalities into shaded regions aids in accurately representing the set.

15–22. Sets in polar coordinates Sketch the following sets of points.2 ≤ r ≤ 8