AMC12 2007 B

AMC12 2007 B · Q19

AMC12 2007 B · Q19. It mainly tests Polygons, 3D geometry (volume).

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

边长为$6$的菱形$ABCD$通过将$\overline{AB}$与$\overline{DC}$粘贴卷成一个体积为$6$的圆柱。求$\sin(\angle ABC)$。

(A) \frac{\pi}{9} \frac{\pi}{9}

(B) \frac{1}{2} \frac{1}{2}

(D) \frac{\pi}{4} \frac{\pi}{4}

(E) \frac{\sqrt{3}}{2} \frac{\sqrt{3}}{2}

Answer

Correct choice: (A)

正确答案：（A）

Solution

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

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

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