Question

如图，从一块圆形饼干面团中切出七块半径为 $1$ 英寸的饼干。相邻饼干相切，除中心饼干外，其他饼干都与面团边缘相切。剩下的边角料被重新塑造成另一块同样厚度的饼干。边角料饼干的半径是多少英寸？

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?

$\sqrt{2}$

$1.5$

$\sqrt{\pi}$

$\sqrt{2\pi}$

$\pi$

答案：A

知识点：圆面积相切圆

难度评级：1140

解答：

圆形面团半径为 $3$ 英寸，因为从中心到边缘经过一个饼干直径再加一个饼干半径。

面团面积为 $3^2\pi = 9\pi$ ，七块饼干的总面积为 $7 \cdot 1^2\pi = 7\pi$ 。

剩余边角料的面积为 $9\pi - 7\pi = 2\pi$ ，所以它的半径为 $\sqrt{2}$ 。

所以正确答案是 A。

The circle of cookie dough has a radius of $3$ inches since it is the same as the diameter plus the radius of a cookie.

The area of the cookie dough is $3^2\pi = 9\pi,$ and the cookies have an area of $7 \cdot 1^2\pi = 7\pi.$

The area of the leftover scrap is therefore $9\pi - 7\pi = 2\pi.$ This means that its radius is $\sqrt{2}.$

Thus, the correct answer is A .

2016 AMC 10A 第 15 题

答案：A

难度评级：1140

解答：

其他年份的第 15 题