AMC10 2009 B

AMC10 2009 B · Q22

AMC10 2009 B · Q22. It mainly tests Similarity, Area & perimeter.

A cubical cake with edge length 2 inches is iced on the sides and the top. It is cut vertically into three pieces as shown in this top view, where M is the midpoint of a top edge. The piece whose top is triangle B contains c cubic inches of cake and s square inches of icing. What is $c + s$?

一个边长 2 英寸的立方体蛋糕在侧面和顶面涂了糖霜。如顶视图所示，它被垂直切成三块，其中 M 是顶边中点。三角形 B 的顶部那块蛋糕包含 $c$ 立方英寸的蛋糕和 $s$ 平方英寸的糖霜。求 $c + s$。

(A) $\frac{24}{5}$ $\frac{24}{5}$

(B) $\frac{32}{5}$ $\frac{32}{5}$

(D) $5 + \frac{16\sqrt{5}}{5}$ $5 + \frac{16\sqrt{5}}{5}$

(E) $10 + 5\sqrt{5}$ $10 + 5\sqrt{5}$

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): The area of triangle $A$ is $1$, and its hypotenuse has length $\sqrt{5}$. Triangle $B$ is similar to triangle $A$ and has a hypotenuse of $2$, so its area is $(\frac{2}{\sqrt{5}})^2=\frac{4}{5}$. The volume of the required piece is $c=\frac{4}{5}\cdot 2=\frac{8}{5}$ cubic inches. The icing on this piece has an area of $s=\frac{4}{5}+2^2=\frac{24}{5}$ square inches. Therefore $c+s=\frac{8}{5}+\frac{24}{5}=\frac{32}{5}$.

答案（B）：三角形$A$的面积是$1$，其斜边长为$\sqrt{5}$。三角形$B$与三角形$A$相似，且斜边为$2$，所以其面积为$(\frac{2}{\sqrt{5}})^2=\frac{4}{5}$。所需那块的体积为$c=\frac{4}{5}\cdot 2=\frac{8}{5}$立方英寸。这块上面的糖霜面积为$s=\frac{4}{5}+2^2=\frac{24}{5}$平方英寸。因此$c+s=\frac{8}{5}+\frac{24}{5}=\frac{32}{5}$。

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