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\noindent {\bf Problem statement} The thread length for a simple spool
of cotton thread is 25 yards. To celebrate Valentine's Day, purchase a
spool of red thread and send it to your beloved with these
instructions:

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\font\bss=cmss10.tfm 
{\narrower
\narrower \noindent \bss Unwind the thread and arrange it in the shape of
a cardioid, $r=A(1-\sin \theta)$. The area of that cardioid represents
how much I love you compared to the ordinary Valentine's Day card!

}

\medskip

\noindent Compute the arclength of 
$r=A(1-\sin \theta)$ and find $A$ so that the length is 25 yards. Then
compute the area inside that cardioid. Sketch the result. 

\medskip

\noindent {\bf Comment} An opened standard greeting card seems to have
area about 70 inches$^2$, or about .054 yards$^2$. On sale, a spool of
thread costs about 25 cents. A card these days costs several
dollars. Isn't the thread more cost-effective?






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