Question

A dimpled limaçon with no inner loop is graphed on a polar coordinate grid.The graph looks like a circle with a small indent on the bottom. Most of the graph is above the pole. The vertical intercepts are located at left-parenthesis 5 comma Start Fraction pi over 2 End Fraction right-parenthesis and left-parenthesis 1 comma Start Fraction 3 pi over 2 End Fraction right-parenthesis. The horizontal intercepts are located at left-parenthesis 3 comma 0 right-parenthesis and left-parenthesis negative 3 comma 0 right-parenthesis.
Question
Use the image to answer the question.
Write the equation for the graph of the limaçon.
(1 point)
Responses

r = 3 + 2 cos θ
r = 3 + 2 cos θ

r = 3 + 2 sin θ
r = 3 + 2 sin θ

r = 2 + 3 cos θ
r = 2 + 3 cos θ

r = 2 + 3 sin θ
r = 2 + 3 sin θ
Skip to navigation

Answer 1

To determine the equation of the limaçon based on your description, let's analyze the provided information:

1. **Vertical intercepts**: You have two vertical intercepts at \( \left(5, \frac{\pi}{2}\right) \) and \( \left(1, \frac{3\pi}{2}\right) \). The first intercept at \( \left(5, \frac{\pi}{2}\right) \) indicates that when \( \theta = \frac{\pi}{2} \), \( r = 5 \), and the second intercept at \( \left(1, \frac{3\pi}{2}\right) \) indicates that when \( \theta = \frac{3\pi}{2} \), \( r = 1 \).

2. **Horizontal intercepts**: The horizontal intercepts at \( (3, 0) \) and \( (-3, 0) \) indicate that when \( \theta = 0 \), \( r = 3 \), and when \( \theta = \pi \), \( r = -3 \).

Given this information, we see that the limaçon crosses the y-axis at values 5 and 1, and it reaches its maximum at \( \theta = 0 \) with a value of 3.

Since there is a dimple on the graph and no inner loop, the equation generally takes the form:
\[
r = a + b \cos(\theta)
\]
or
\[
r = a + b \sin(\theta)
\]
where \( a \) and \( b \) determine the shape of the limaçon.

Given the intercepts:
- The maximum \( r \) at \( \theta = 0 \) suggests that for \( r = 3 + 2 \cos(\theta) \), the points would yield the correct intercepts.

To summarize:
- \( r = 3 \) when \( \theta = 0 \).
- \( r = 5 \) when \( \theta = \frac{\pi}{2} \).
- \( r = 1 \) when \( \theta = \frac{3\pi}{2} \).

This aligns with:
\[
r = 3 + 2 \cos(\theta)
\]

Thus, the equation that matches the description of the limaçon with the given intercepts is:

**Response:**
\( r = 3 + 2 \cos(\theta) \)