Difference between revisions of "2007 AMC 12B Problems/Problem 19"

Revision as of 00:19, 7 February 2011

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 $\sin(\angle ABC)$ ?

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

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]$

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

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

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

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

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

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

2007 AMC 12B (Problems • Answer Key • 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
All AMC 12 Problems and Solutions

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

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Difference between revisions of "2007 AMC 12B Problems/Problem 19"

Revision as of 00:19, 7 February 2011

Problem 19

Solution

See Also