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Polar Art Project

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Polar Art Project
Graph of r = b sin (kθ) when k = 2 (even number), for varying values of b = 1, 2, 3, 6

Instructions: The major point of this project is that it convincingly communicate what you have found to be important from your investigation. Your write-up should be typed, double spaced, and include graphs (embedded in the document). Use complete sentences. Clarity, accurate mathematics, terminology and correctness of responses will determine the grade. You can use Desmos, Fooplot or Wolframalpha for these investigations.

Investigate graphs of the polar equations r = b cos (kθ) and r = b sin (kθ) for the following two cases:

• Case I. For b = 1, 2, 3, 6 when k = even number
• Case II. For b = 1, 2, 3, 6 when k = odd number

Note: Try different even & odd numbers to ascertain that the conclusion is correct.

In your write-up do make sure to address the following questions:

a. How many leaves do the resulting roses have? Are the leaves equal in size? What is the relationship between "k" and the number of leaves?
b. How does varying "b" affect the graph?
c. What can you say about the symmetry of the graphs?

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Solution Summary

The solution contains 7 pages, with 373 words and 8 figures. Each figure contains 4 polar curves (for a total of 32 polar curves). The solution has been separated into sections for Case I and Case II. A summary of findings is provided at the end of each case.

Solution Preview

This project is about the shapes of a type of polar curve known as a rose. The rose is also known as a rhodenea curve, and is represented by a sinusoidal equation and is plotted in polar coordinates.

The general equation of a rose is r = b cos (kθ) or r = b sin (kθ). The values of b and k can be varied to produce different numbers of leaves and orientations. Several different cases are shown in the next few pages. A summary that answers the ...

Solution provided by:
J Pye, MSc
Education
  • MSc, California State Polytechnic University, Pomona
  • MBA, University of California, Riverside
  • BSc, California State Polytechnic University, Pomona
  • BSc, California State Polytechnic University, Pomona
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