Question

一块均匀密度的薄木板是边长为 $3$ 英寸的等边三角形，重 $12$ 盎司。另一块同种木材、同样厚度的等边三角形木板边长为 $5$ 英寸。下面哪个数最接近第二块木板的重量（盎司）？

A thin piece of wood of uniform density in the shape of an equilateral triangle with side length $3$ inches weighs $12$ ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has side length $5$ inches. Which of the following is closest to the weight, in ounces, of the second piece?

$\ 14.0$

$\ 16.0$

$\ 20.0$

$\ 33.3$

$\ 55.6$

答案：D

知识点：相似长度、面积与体积的缩放关系

难度评级：960

解答：

面积按 $\frac 53 ^2$ 倍增加，因为边长按 $\frac 53$ 倍增加。厚度不变，所以体积也按 $\frac 53^2$ 倍增加。新重量为约为 $33.3$ 。 $12 \cdot \dfrac {25}{9} = \dfrac{100}3.$

所以正确答案是 D。

The surface area is increased by a factor of $\frac 53 ^2$ since the side lengths are increased by a factor of $\frac 53.$ Also since the thickness is constant, the volume is scaled up by that much. Then, the wood having the same density makes the weight increase by a factor of $\frac 53^2$ as well, so the new weight is $12 \cdot \dfrac {25}{9} = \dfrac{100}3.$ This is approximately $33.3.$

Thus, the correct answer is D .

2016 AMC 10B 第 10 题

答案：D

难度评级：960

解答：

其他年份的第 10 题