Question

Abdul 和 Chiang 站在一片田野中，相距 $48$ 英尺。Bharat 也站在同一片田野中，并且在使他看向 Abdul 和 Chiang 的两条视线所成角为 $60^\circ$ 的条件下，尽可能远离 Abdul。Abdul 与 Bharat 之间距离的平方是多少（单位为平方英尺）？

Abdul and Chiang are standing $48$ feet apart in a field. Bharat is standing in the same field as far from Abdul as possible so that the angle formed by his lines of sight to Abdul and Chiang measures $60^\circ.$ What is the square of the distance (in feet) between Abdul and Bharat?

$1728$

$2601$

$3072$

$4608$

$6912$

答案：C

知识点：圆周角正弦定理最优化

难度评级：1590

解答：

设 $A$ 为 Abdul， $C$ 为 Chiang，且 $AC = 48$ ， $B$ 为 Bharat，满足 $\angle B = 60^\circ$ 。所有以 $60^\circ$ 角看见 $AC$ 的点都在同一圆弧上，所以所有可行的 $B$ 都在一个圆上，其中弦 $AC$ 所对圆周角为 $60^\circ$ 。正弦定理给出该圆直径为 $\frac{AC}{\sin 60^\circ} = \frac{48}{\sqrt3/2} = 32\sqrt3$ 。现在 $AB$ 是一条弦，而弦最长时为直径。所以 $AB = 32\sqrt3$ ， $AB^2 = 1024 \cdot 3 = 3072$ 。因此，正确答案是 C。

Let $A$ be Abdul, $C$ be Chiang with $AC = 48,$ and $B$ be Bharat with $\angle B = 60^\circ.$ Every point seeing $AC$ at $60^\circ$ lies on one circular arc, so all valid $B$ sit on a circle where chord $AC$ subtends $60^\circ.$ The law of sines gives its diameter, $\frac{AC}{\sin 60^\circ} = \frac{48}{\sqrt3/2} = 32\sqrt3.$ Now $AB$ is a chord, and a chord is longest when it's a diameter. So $AB = 32\sqrt3$ and $AB^2 = 1024 \cdot 3 = 3072.$ Thus, C is the correct answer.

2023 AMC 10A 第 13 题

答案：C

难度评级：1590

解答：

其他年份的第 13 题