1956 AMC 12 第 37 题

Question

一幅地图的比例尺为一英寸半表示

400

英里。图上某庄园呈一个含有

60^\circ

角的菱形，

60^\circ

角所对的对角线长为

\dfrac3{16}

英寸。该庄园的实际面积为多少平方英里？

On a map whose scale is

400

miles to an inch and a half, a certain estate is represented by a rhombus having a

60^\circ

angle. The diagonal opposite

60^\circ

is

\dfrac3{16}

in. The area of the estate in square miles is:

\dfrac{2500}{\sqrt3}

\dfrac{1250}{\sqrt3}

1250

\dfrac{5625\sqrt3}{2}

1250\sqrt3

在含有

60^\circ

角的菱形中，较短的对角线等于边长

In a

60^\circ

rhombus, the shorter diagonal equals the side length

比例尺为每英寸

\frac{800}{3}

英里，所以先把

\frac{3}{16}

英寸的边长换算为实际长度，再求面积

The scale is

\frac{800}{3}

miles per inch, so convert the

\frac{3}{16}

-inch side before finding area

60^\circ

角所对的对角线把菱形分成两个等边三角形，所以它的长度等于菱形的边长。地图的比例尺为每英寸

\frac{800}{3}

英里，因此实际边长为

\frac3{16}\cdot\frac{800}{3}=50

英里。所以菱形的面积为

\begin{aligned} 50^2\sin60^\circ &=2500\cdot\frac{\sqrt3}{2}\\ &=1250\sqrt3 \end{aligned}\text{。}

因此，正确答案是 E。

The diagonal opposite the

60^\circ

angle divides the rhombus into two equilateral triangles, so its length equals the rhombus side. The map scale is

\frac{800}{3}

miles per inch, hence the actual side length is

\frac3{16}\cdot\frac{800}{3}=50

miles. Therefore the rhombus area is

\begin{aligned} 50^2\sin60^\circ &=2500\cdot\frac{\sqrt3}{2}\\ &=1250\sqrt3. \end{aligned}

Thus, the correct answer is E.

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