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2007 AMC 12B Problems/Problem 19

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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

$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("...$

$V_{Cylinder} = \pi r^2 h$

Where $C = 2\pi r = 6$ and $h=6\sin\theta$

$r = \frac{3}{\pi}$

$V = \pi \left(\frac{3}{\pi}\right)^2\cdot 6\sin\theta$

$6 = \frac{9}{\pi} \cdot 6\sin\theta$

$\sin\theta = \frac{\pi}{9} \Rightarrow \mathrm{(A)}$

See Also

2007 AMC 12B (Problems • Resources)
Preceded by Problem 18	Followed by Problem 20
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25

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