# 2007 AMC 12B Problems/Problem 19

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## Problem

Rhombus $ABCD$, with side length $6$, is rolled to form a cylinder of volume $6$ by taping $\overline{AB}$ to $\overline{DC}$. What is $\sin(\angle ABC)$? $\mathrm{(A)}\ \frac{\pi}{9} \qquad \mathrm{(B)}\ \frac{1}{2} \qquad \mathrm{(C)}\ \frac{\pi}{6} \qquad \mathrm{(D)}\ \frac{\pi}{4} \qquad \mathrm{(E)}\ \frac{\sqrt{3}}{2}$

## Solution $[asy] pair B=(0,0), A=(6*dir(60)), C=(6,0); pair D=A+C; draw(A--B--C--D--A); draw(A--(3,0)); label("$$A$$",A,NW);label("$$B$$",B,SW);label("$$C$$",C,SE);label("$$D$$",D,NE); label("$$6$$",A/2,NW); label("$$\theta$$",(.8,.5)); label("$$h$$",(3,2.6),E); [/asy]$ $V_{\mathrm{Cylinder}} = \pi r^2 h$

Where $C = 2\pi r = 6$ and $h=6\sin\theta$ $r = \frac{3}{\pi}$ $V = \pi \left(\frac{3}{\pi}\right)^2\cdot 6\sin\theta$ $6 = \frac{9}{\pi} \cdot 6\sin\theta$ $\sin\theta = \frac{\pi}{9} \Rightarrow \mathrm{(A)}$

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