1955 AMC 12 第 15 题

Question

两个同心圆的面积之比为

1:3

。若较小圆的半径为

r

，两圆半径之差最接近：

The ratio of the areas of two concentric circles is

1:3.

If the radius of the smaller is

r,

then the difference between the radii is best approximated by:

0.41r

0.73

0.75

0.73r

0.75r

面积之比等于半径之比的平方

Area ratios are the squares of radius ratios

较大圆的半径为

r\sqrt3

，所以估算

\sqrt3-1

The larger radius is

r\sqrt3

, so estimate

\sqrt3-1

若较大圆的半径为

R

，则

\frac{\pi r^2}{\pi R^2}=\frac13\text{，}

所以

R=r\sqrt3

。两半径之差为

R-r=(\sqrt3-1)r\approx0.732r\text{。}

因此，最佳近似值为

0.73r

，正确答案是 D。

If the larger radius is

R,

then

\frac{\pi r^2}{\pi R^2}=\frac13,

so

R=r\sqrt3.

The difference is

R-r=(\sqrt3-1)r\approx0.732r.

Thus, the best approximation is

0.73r,

and the correct answer is D.

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