Question
3. Identify the graph of the polar equation, do not draw.. \( r=7 \cos (4 \theta) \)
Ask by Olson Gardner. in the United States
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The graph of the equation \( r = 7 \cos(4\theta) \) is an 8-petaled rose curve, with each petal extending up to 7 units from the origin.
Solution
1. Recognize the general form of a rose curve in polar coordinates:
\[
r = a \cos(k\theta) \quad \text{or} \quad r = a \sin(k\theta)
\]
2. Identify the parameters in the given equation:
\[
r = 7 \cos(4\theta)
\]
Here, \(a = 7\) and \(k = 4\).
3. Determine the number of petals:
- If \(k\) is even, a rose curve has \(2k\) petals.
- Since \(k = 4\) is even, the curve has \(2 \times 4 = 8\) petals.
4. Note the length (or radius) of the petals is determined by \(a\):
Each petal extends to a maximum distance of \(7\) units from the origin.
Thus, the graph of the equation
\[
r = 7 \cos(4\theta)
\]
is an 8-petaled rose curve with each petal reaching up to a length of \(7\).
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Simplify this solution Beyond the Answer
This polar equation describes a rose curve with petals! Specifically, since the coefficient of \(\theta\) is 4, it will have 8 petals (double the coefficient) and will be symmetric about the x-axis. The petals will extend out to a maximum radius of 7, which means they’ll stretch out to 7 units from the origin at their widest points. In real-world applications, rose curves can be found in various fields, from art to nature; patterns resembling rose curves often appear in flower shapes or in the design of certain structures. Knowing how to identify such curves can help in fields like robotics for path planning, or even in graphic design for creating visually appealing motifs!
