AMC10 2016 A

AMC10 2016 A · Q15

AMC10 2016 A · Q15. It mainly tests Circle theorems, Area & perimeter.

Seven cookies of radius 1 inch are cut from a circle of cookie dough, as shown. Neighboring cookies are tangent, and all except the center cookie are tangent to the edge of the dough. The leftover scrap is reshaped to form another cookie of the same thickness. What is the radius in inches of the scrap cookie?

如图所示，从一块圆形的饼干面团中切出 7 块半径为 1 英寸的圆形饼干。相邻的饼干互相相切，且除中心那块饼干外，其余饼干都与面团的外边缘相切。剩余的边角料被重新塑形，做成另一块厚度相同的饼干。问这块“边角料饼干”的半径是多少英寸？

(A) $\sqrt{2}$ $\sqrt{2}$

(B) 1.5 1.5

(D) $\sqrt{2\pi}$ $\sqrt{2\pi}$

(E) $\pi$ $\pi$

Answer

Correct choice: (A)

正确答案：（A）

Solution

Answer (A): The circle of dough has radius 3 inches. The area of the remaining dough is $3^2\cdot\pi-7\pi=2\pi$ in$^2$. Let $r$ be the radius in inches of the scrap cookie; then $2\pi=\pi r^2$. Therefore $r=\sqrt{2}$ inches.

答案（A）：这块面团是半径为 3 英寸的圆。剩余面团的面积为 $3^2\cdot\pi-7\pi=2\pi$ 平方英寸。设边角料饼干的半径为 $r$（英寸），则 $2\pi=\pi r^2$。因此 $r=\sqrt{2}$ 英寸。

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