Let's examine the relationship between the coefficients [tex]\(a\)[/tex] and [tex]\(b\)[/tex] in the equation of a limaçon and how this relationship affects the presence of an inner loop.
The general form of a limaçon is given by:
[tex]\[ r = a + b \cos(\theta) \][/tex]
For the limaçon in the given problem:
[tex]\[ r = 3 + 4 \cos(\theta) \][/tex]
we can identify that [tex]\( a = 3 \)[/tex] and [tex]\( b = 4 \)[/tex].
To determine the relationship between [tex]\(a\)[/tex] and [tex]\(b\)[/tex] that dictates the presence of an inner loop, we need to analyze the quotient [tex]\(\frac{a}{b}\)[/tex] and [tex]\(\frac{b}{a}\)[/tex].
First, compute the quotient [tex]\(\frac{a}{b}\)[/tex]:
[tex]\[ \frac{a}{b} = \frac{3}{4} = 0.75 \][/tex]
Next, compute the quotient [tex]\(\frac{b}{a}\)[/tex]:
[tex]\[ \frac{b}{a} = \frac{4}{3} \approx 1.333 \][/tex]
Now, we need to use these quotients to determine the nature of the limaçon. For a limaçon:
- If [tex]\(\frac{a}{b} > 1\)[/tex], the limaçon has no inner loop and is a dimpled limaçon.
- If [tex]\(\frac{b}{a} > 1\)[/tex], the limaçon has an inner loop.
Given the values we obtained:
- [tex]\(\frac{a}{b} = 0.75\)[/tex], which is less than 1.
- [tex]\(\frac{b}{a} = 1.333\)[/tex], which is greater than 1.
Since [tex]\(\frac{b}{a} > 1\)[/tex], this indicates that the curve is a limaçon with an inner loop.
Hence, the correct statement is:
[tex]\[ \text{Because } \frac{b}{a} > 1, \text{ the curve is a limaçon with an inner loop.} \][/tex]
