Problem 27 - Essential Calculus

Question

Convert the polar equation to rectangular form and sketch its graph. $$ r=3 $$

Step-by-Step Solution

Answer

The equation in rectangular form is \(x^{2} + y^{2} = 9\) and its graph is a circle centered at origin with radius of 3 units.

Step 1: Convert to Rectangular Form

To convert the polar equation \(r=3\) to rectangular coordinates, we can use the relationship \(r^{2}=x^{2}+y^{2}\). Since \(r=3\), we square both sides to obtain \(r^{2}=9\). Replacing \(r^{2}\) by \(x^{2} + y^{2}\) gives us the equation \(x^{2} + y^{2}=9\).

Step 2: Sketch the Graph

The equation \(x^{2} + y^{2}=9\) represents a circle in the rectangular coordinate system. The circle is centered at the origin (0,0) and has a radius of 3 units. So, when sketching, start at the origin and draw a circle that extends 3 units in all directions.