Problem 67 - Essential Calculus

Question

Sketch a graph of the polar equation. $$ r=5 $$

Step-by-Step Solution

Answer

The graph of the polar equation \( r = 5 \) is a circle with a radius of 5 units, centered at the origin, on the polar axis.

Step 1: Understand the basics of the polar coordinate system

A polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. Here, the reference point is the origin (0,0), and the reference direction is the positive x-axis. The polar coordinates (r,θ) are then given by the radius r and the angle θ.

Step 2: Translate the polar equation to corresponding graph representation

The given equation is \( r = 5 \), which describes a circle of radius 5 centered at the origin. There is no θ term in the equation, which means that the circle is not rotated - it lies in the standard position, centered at the origin.

Step 3: Sketch the graph

Draw the polar axis, which is a horizontal line called the initial line. Choose a length for 5 units on the paper and mark it on the initial line. Draw a circle with radius 5 units centered at the origin. This circle represents the graph of the polar equation \( r = 5 \).