Question

一根铁丝被剪成两段，长度分别为 $a$ 和 $b$ 。长度为 $a$ 的一段弯成等边三角形，长度为 $b$ 的一段弯成正六边形。三角形和六边形面积相等。 $\frac{a}{b}$ 是多少？

A wire is cut into two pieces, one of length $a$ and the other of length $b.$ The piece of length $a$ is bent to form an equilateral triangle, and the piece of length $b$ is bent to form a regular hexagon. The triangle and the hexagon have equal area. What is $\frac{a}{b}?$

$1$

$\dfrac{\sqrt{6}}{2}$

$\sqrt{3}$

$2$

$\dfrac{3\sqrt{2}}{2}$

答案：B

知识点：正多边形等边三角形长度、面积与体积的缩放关系

难度评级：1420

解答：

设等边三角形边长为 $s$ 。面积为 $A$ 。则 $a=3s$ 。

边长为 $s$ 的正六边形由 $6$ 个边长为 $s$ 的等边三角形组成，面积为 $6A$ 。

若正六边形面积也为 $A$ ，它的边长应为 $\dfrac{s}{\sqrt 6}$ ，所以 $b = 6\cdot \dfrac{s}{\sqrt 6} = s\sqrt 6 .$

因此 $\dfrac ab = \dfrac{3s}{s\sqrt 6} = \dfrac{3 \sqrt 6}6 = \dfrac{\sqrt 6} 2.$

所以正确答案是 B。

Let the side length of the equilateral triangle be $s.$ Then, let its area be $A.$ This would make $a=3s.$

As such, a hexagon with side $s$ would have $6$ equilateral triangles, with side length $s,$ making its area $6A.$

Therefore, the side length of the hexagon with area $A$ is equal to $\dfrac{s}{\sqrt 6} .$ As such, $b = 6\cdot \dfrac{s}{\sqrt 6} = s\sqrt 6 .$

This makes $\dfrac ab = \dfrac{3s}{s\sqrt 6} = \dfrac{3 \sqrt 6}6 = \dfrac{\sqrt 6} 2.$

Thus, the correct answer is B .

2013 AMC 10B 第 15 题

答案：B

难度评级：1420

解答：

其他年份的第 15 题