Question

一个大正方形区域铺有 $n^2$ 块阴影正方形瓷砖，每块边长为 $s$ 英寸。每块瓷砖周围有宽 $d$ 英寸的边框。下图显示了 $n=3$ 的情况。当 $n=24$ 时， $576$ 块阴影瓷砖覆盖了大正方形区域面积的 $64\%$ 。当 $n$ 取这个较大的值时， $\frac{d}{s}$ 是多少？

A large square region is paved with $n^2$ shaded square tiles, each measuring $s$ inches on a side. A border $d$ inches wide surrounds each tile. The figure below shows the case for $n=3.$ When $n=24,$ the $576$ shaded tiles cover $64\%$ of the area of the large square region. What is the ratio $\frac{d}{s}$ for this larger value of $n?$

$\dfrac{6}{25}$

$\dfrac{1}{4}$

$\dfrac{9}{25}$

$\dfrac{7}{16}$

$\dfrac{9}{16}$

答案：A

知识点：面积比正方形（几何）

难度评级：1510

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文字解答：

当 $n=24$ 时，阴影瓷砖总面积为 $24^2s^2$ 大正方形一边有 $24$ 块瓷砖和 $25$ 条边框，所以边长为 $24s+25d$ 。

阴影部分占 $64\%=\dfrac{16}{25}$ ，所以取正平方根得 $\dfrac{24s}{24s+25d}=\dfrac{4}{5}$ 。 $\dfrac{24^2s^2}{(24s+25d)^2}=\dfrac{16}{25}.$

因此 $120s=96s+100d$ ，所以 $24s=100d$ ， $\dfrac{d}{s}=\dfrac{6}{25}$ 。

正确答案是 A。

For $n=24,$ the shaded tile area is $24^2s^2.$ Each side of the large square consists of $24$ tiles and $25$ borders, so its side length is $24s+25d.$

The shaded tiles cover $64\%=\dfrac{16}{25}$ of the large square, so $\dfrac{24^2s^2}{(24s+25d)^2}=\dfrac{16}{25}.$ Taking positive square roots gives $\dfrac{24s}{24s+25d}=\dfrac{4}{5}.$

Thus $120s=96s+100d,$ so $24s=100d$ and $\dfrac{d}{s}=\dfrac{6}{25}.$

Thus, the correct answer is A.

2020 AMC 8 第 24 题

答案：A

难度评级：1510

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