Question

一对标准的 $6$ -面均匀骰子掷一次。掷出的点数和决定一个圆的直径。圆的面积数值小于圆周长数值的概率是多少？

A pair of standard $6$ -sided fair dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle’s circumference?

$\dfrac{1}{36}$

$\dfrac{1}{12}$

$\dfrac{1}{6}$

$\dfrac{1}{4}$

$\dfrac{5}{18}$

答案：B

知识点：骰子（概率）圆面积不等式

难度评级：1140

小提示：

比较 $\pi(\frac{d}{2})^2$ 和 $\pi d$ 。

Compare $\pi(\frac{d}{2})^2$ with $\pi d$

大提示：

不等式化为 $d<4$ 。

The inequality reduces to $d<4$

解答：

要使面积小于周长，必须有 $\pi r^2 \lt 2\pi r$ $r \lt 2\text{。}$

因此直径必须小于 $4$ 。满足条件的掷法有三种： $(1, 1), (1, 2), (2, 1)\text{。}$

因此概率为 $\dfrac{3}{36} = \dfrac{1}{12}\text{。}$

所以正确答案是 B。

For the area to be less than the circumference, we must have $\pi r^2 \lt 2\pi r$ $r \lt 2.$

This means the diameter must be less than $4.$ There are three possible rolls that satisfy this: $(1, 1), (1, 2), (2, 1).$

The probability is then $\dfrac{3}{36} = \dfrac{1}{12}.$

Thus, B is the correct answer.

2011 AMC 10A 第 14 题

答案：B

难度评级：1140

小提示：

大提示：

解答：

其他年份的第 14 题