Question

match each graph with its equation
a. $r = 5 + 3\cos(\theta)$
b. $r = 3\cos(\theta)$
c. $r = 2 + 3\cos(\theta)$
d. $r = 7 + 3\cos(\theta)$
e. $r = 3 + 3\cos(\theta)$

Explanation:

Step1: Analyze the general form of polar equations

The polar equation of a limacon is \(r = a + b\cos(\theta)\). When \(a>b\), it is a convex limacon; when \(a = b\), it is a cardioid; when \(a < b\), it has an inner loop.

Step2: Analyze each equation

• For \(r = 3\cos(\theta)\), it is a circle. The standard form of a circle in polar coordinates \(r = 2R\cos(\theta)\) (centered at \((R,0)\) with radius \(R\)). Here \(R=\frac{3}{2}\).
• For \(r = 2+3\cos(\theta)\), since \(a = 2\), \(b = 3\) and \(a < b\), it has an inner loop.
• For \(r=3 + 3\cos(\theta)\), since \(a = 3\), \(b = 3\) and \(a=b\), it is a cardioid.
• For \(r = 5+3\cos(\theta)\) and \(r = 7+3\cos(\theta)\), since \(a>b\) (\(a = 5\) or \(a = 7\), \(b = 3\)), they are convex limacons. The larger \(a\) is, the larger the limacon.

Answer:

The graph with an inner loop matches \(c. r = 2+3\cos(\theta)\), the circle matches \(b. r = 3\cos(\theta)\)