Question

$A$ 和 $B$ 从圆形跑道上一对直径相对的点同时出发，以匀速沿相反方向绕跑道行进。他们在 $B$ 行进 $100$ 码后第一次相遇，并在 $A$ 距离完成一圈还有 $60$ 码时第二次相遇。则跑道的周长（单位为码）是：

$A$ and $B$ travel around a circular track at uniform speeds in opposite directions, starting from diametrically opposite points. If they start at the same time, meet first after $B$ has travelled $100$ yards, and meet a second time $60$ yards before $A$ completes one lap, then the circumference of the track in yards is:

$400$

$440$

$480$

$560$

$880$

答案：C

知识点：相对速度圆周长比与比例方程组

难度评级：2190

小提示：

设跑道周长为 $2C$ ，比较两次相遇时两人的路程之比

Let the circumference be $2C$ and compare the two runners’ traveled-distance ratios at each meeting

大提示：

第一次相遇时两人的路程为 $C-100$ 和 $100$ ；第二次相遇时为 $2C-60$ 和 $C+60$

At the first meeting their distances are $C-100$ and $100$ ; at the second they are $2C-60$ and $C+60$

解答：

设跑道周长为 $2C$ 。第一次相遇时， $B$ 已行进 $100$ 码， $A$ 已行进 $C-100$ ，所以他们的速度之比为 $\frac{v_A}{v_B}=\frac{C-100}{100}\text{。}$ 第二次相遇时， $A$ 距离完成一圈还差 $60$ 码，所以它已行进 $2C-60$ 。此时两人的路程之和为 $3C$ ，所以 $B$ 已行进 $C+60$ 。因此 $\frac{C-100}{100}=\frac{2C-60}{C+60}\text{。}$ 交叉相乘可得 $C=240$ ，所以跑道周长为 $2C=480$ 码。

因此，正确答案是 C。

Let the circumference be $2C.$ At the first meeting, $B$ has traveled $100$ yards and $A$ has traveled $C-100,$ so their speed ratio is $\frac{v_A}{v_B}=\frac{C-100}{100}.$ At the second meeting, $A$ is $60$ yards short of one lap, so it has traveled $2C-60.$ By then their combined distance is $3C,$ so $B$ has traveled $C+60.$ Hence $\frac{C-100}{100}=\frac{2C-60}{C+60}.$ Cross-multiplying gives $C=240,$ so the circumference is $2C=480$ yards.

Therefore, the correct answer is C.

1970 AMC 12 第 32 题

答案：C

难度评级：2190

小提示：

大提示：

解答：

其他年份的第 32 题