Dear Calculus Student,

You must be the Lone Math Ranger, the way you swooped in and solved those nasty equations! I owe you my very life, and I won't forget it. I don't know if the unstoppable Officer Kovaleskia told you or not, but they found Phaze's fingerprints on the note that was sent with the mugs (well-preserved under a piece of tape&emdash;what a doofus). That evidence, together with your incredible mathematical savvy, has sealed the prosecution's case against him. Phaze is in the hoosegow now! Who was that masked mathematician?!?

My doctor here at the Big City Lady of Pity Hospital is wonderful, too: she's Jonas Salk and Florence Nightingale rolled up into one. She says that the information you passed along about my bones made all the difference in my treatment. She's spent a lot of time working with me, and I feel so much better now&emdash;and the reason for my good spirits isn't entirely medical, I must admit. Dr. Clementine (Darlene, to her friends) is a real winner. She's smart, she's athletic, she's pretty&emdash;and she even likes grapefruits! Her only blemish, if I dare call it that, is that she lacks depth perception. Still, she's managed to overcome that handicap better than any depth-perception-challenged person I know. I can't think of anyone I'd rather share the rest of my life with, and I'm going to do my best to convince her to feel the same way about me.

So since you're being such a studly math-is-no-problem kind of a hero, maybe you can help me one last time. Darlene has been on a real calculus kick ever since some math students showed her track team how to win the 400-meter relay. Her teammate Betty lost her heart to a guy who was good at derivatives and poetry both, and Darlene says that she too is saving her love for the man who can solve her problem challenge. (It sounds like a princess setting a task for a knight, but I'd do anything she would tell me to).

She has a white-gold medallion in the shape of a circle, exactly 2 cm across. She wants to inlay in this medallion a flower made of yellow gold&emdash;but not just any flower: a rose curve. And she needs to know how much yellow gold she'll need. So I tried: a 3-petal rose curve r = sin(3q) has area p /4 cm2. (I got this by calculating that the area of each of its petals is 1/2º0 p/3 r^2 dq). The area of a 5-petal rose curve r = sin(5q) is also p /4 cm2. But the area of a 4-petal rose curve r = sin(2q) is p /2 cm2 .

I proudly explained this to my darling Darlene Clementine, but she just laughed: "So what's the area of a 6-petal rose curve?" I shot back, "p/2 cm2! . . . wait . . . the curve r = sin(6q) has 12 petals and r = sin(3q) has 3 petals . . . there is no such thing as a 6-petal rose curve!"

She laughed that silvery laugh again and said, "Check out r = sin(3/2q)". Lo and behold&emdash;I think she may be right! But I have no idea how to get the area of the complicated picture I came up with. I'm hoping you can help me!

Even if I come up with the area of this rose curve, I know she'll come back with more complicated ones. Can you pick a beautiful rose curve, one that would be worthy of my darling Darlene Clementine, and tell me how to find how much gold we'll need for the inlay?

I can just see the Marquis sign now: "Greatful cowpoke gets the girl as mathematician rides off into the sunset". This one last favor, Lone Math Ranger, is all I'm asking of you.

Your humble,