1. BY: MOLLY PELAK The Balloon Rose

2. Background Knowledge- -The balloon rose was originally named rhondena meaning “rose”. -It was named after the Italian mathematician, Guido Grandi between 1723 and 1728. -The balloon rose has some interesting properties that we will be discovering today!

3. Properties- -The model equation we will be using today is as follows: r= k+ cos n ∙ θ -Some properties include: - Whatever the value of “n” is, that will be the value for the amount of petals on your rose. - -ex: - If we have the equation of r= 2 + cos 3∙θ, then the rose would have 3 petals as shown in the graph to the top left. - On the other hand, if we have r=2+ cos 4∙θ, we will have a rose with 4 petals also shown in the graph to the bottom left.

4. -While integrating to find the area of our balloon rose with r = 2 +cos 3∙θ, we find another interesting property. -So, we integrate: -We start to see the pattern as we integrate and change the value of “n”. - As “n” values change (# of petals), the area stays the same. - This, too, is another property of the balloon rose!

5. How do we go about integrating?

6. Questions, Comments??? Sources: http://www-history.mcs.st- andrews.ac.uk/PictDisplay/Grandi. html