2011 AMC 10A 第 14 题

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：

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