AMC12 2003 B

AMC12 2003 B · Q21

AMC12 2003 B · Q21. It mainly tests Trigonometry (basic), Geometric probability (basic).

An object moves $8$ cm in a straight line from $A$ to $B$, turns at an angle $\alpha$, measured in radians and chosen at random from the interval $(0,\pi)$, and moves $5$ cm in a straight line to $C$. What is the probability that $AC < 7$?

一个物体沿直线从 $A$ 移动 $8$ cm 到 $B$，然后转一个角度 $\alpha$（以弧度为单位，从区间 $(0,\pi)$ 中随机选择），再沿直线移动 $5$ cm 到 $C$。$AC < 7$ 的概率是多少？

(A) \dfrac{1}{6} \dfrac{1}{6}

(B) \dfrac{1}{5} \dfrac{1}{5}

(D) \dfrac{1}{3} \dfrac{1}{3}

(E) \dfrac{1}{2} \dfrac{1}{2}

Answer

Correct choice: (D)

正确答案：（D）

Solution

By the Law of Cosines, \begin{align*} AB^2 + BC^2 - 2 AB \cdot BC \cos \alpha = 89 - 80 \cos \alpha = AC^2 &< 49\\ \cos \alpha &> \frac 12\\ \end{align*} It follows that $0 < \alpha < \frac {\pi}3$, and the probability is $\frac{\pi/3}{\pi} = \boxed{\textbf{(D) } \frac13 }$.

由余弦定理， \begin{align*} AB^2 + BC^2 - 2 AB \cdot BC \cos \alpha = 89 - 80 \cos \alpha = AC^2 &< 49\\ \cos \alpha &> \frac 12\\ \end{align*} 因此 $0 < \alpha < \frac {\pi}3$，所求概率为 $\frac{\pi/3}{\pi} = \boxed{\textbf{(D) } \frac13 }$。

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