2007 AMC 12B Problems/Problem 19

Problem 19

Rhombus $ABCD$ , with side length $6$ , is rolled to form a cylinder of volume $6$ by taping $\overline{AB}$ to $\overline{DC}$ . What is $\sin(\angle ABC)$ ?

$\mathrm {(A)} \frac{\pi}{9}$ $\mathrm {(B)} \frac{1}{2}$ $\mathrm {(C)} \frac{\pi}{6}$ $\mathrm {(D)} \frac{\pi}{4}$ $\mathrm {(E)} \frac{\sqrt{3}}{2}$

Solution

$[asy] pair A=(0,0), B=(6*dir(60)), D=(6,0); pair C=B+D; draw(A--B--C--D--A); draw(B--(3,0)); label("$A$",A,SW);label("$B$",B,NW);label("$C$",C,NE);label("$D$",D,SE); label("$6$",B/2,NW); label("$\theta$",(.8,.5)); label("$h$",(3,2.6),E); [/asy]$